Vibrations of an alpine ski under structural randomness

In alpine skiing the vibrations of the skis influence the performance of a skier considerably. An alpine ski can be modelled as an Euler‐Bernoulli beam with which the vibrations of the ski can be analysed mathematically. The main characterising property of a ski is its bending stiffness, which can be measured in the lab. Since a ski is a composite material the measured bending stiffness shows variations. These variations can be taken into account by adding a random field to an average bending stiffness. An experiment of a clamped ski is investigated theoretically.


Introduction
In alpine skiing the vibrations of the skis influence the performance of a skier considerably. For theoretical investigations an alpine ski can be modelled as an Euler-Bernoulli beam, where the vibrations are described by the equation of motion of a damped Euler-Bernoulli beam [1] (EI(x)w xx ) xx + ρA(x)w tt + β (EI(x)(w t ) xx ) xx + αρA(x)w t = f.
Here, x denotes the longitudinal coordinate, t the time, EI(x) the bending stiffness, w(x, t) the deflection of the ski in vertical direction, ρ the mass density, A(x) the cross sectional area of the beam, and f the line forces acting on the ski (e.g. the ski-snow interaction). The coefficients β and α stand for the damping coefficients. The main characterising property of a ski is its bending stiffness EI(x), which can be measured in the lab. A ski is a composite material that consists of different layers, which are glued together. Consequently, measurements show variations in the bending stiffness EI(x) in longitudinal direction.
In this work we developed an approach to study the vibrations of an alpine ski, taking into account material variability.

Methods
The variations in the bending stiffness EI(x) of an alpine ski can be considered by adding a random field r(x) to a mean bending stiffness EI 0 (x) in the beam equation (1) A random field assigns a random variable at every point x. The random field r(x) is assumed Gaussian and stationary. Thus, it is completely specified by the constant mean value µ r = E(r(x)) = 0 and the autocorrelation function, taken here as with the so-called correlation length l for any two points x andx [2]. The random field r(x) is simulated based on the Karhunen-Loève expansion [3]. In the following, we describe the solution of (1) with a single realisation of the random bending stiffness EI(x). The vibrations of a ski are investigated in the following situation. The ski is clamped in the centre, the ski tip is deflected at x = L by a load P and released. This experiment accords with the vibrations of a deflected cantilever beam and is modelled by equation (1) with boundary conditions (clamped -free) and initial deflection and velocity: w(x, 0) = w 0 (x), ∂w(x, t) ∂t The initial deflection w 0 (x) is calculated by the static beam equation 2 of 2 Section 5: Nonlinear oscillations and boundary conditions (4). Here, δ represents the Dirac delta function. The differential equations for the static beam equation (6) and the damped Euler-Bernoulli beam (1) are solved by the spectral method [4] using the Ansatz with vectors b(t) = (b k (t)) k=1...N and ϕ(x) = (ϕ k (x)) k=1...N for a given number N . The so-called base functions ϕ k (x) are chosen as the normal modes of a homogeneous cantilever beam, so that the boundary conditions (4) are fulfilled. Consequently, the spectral method yields the system of linear ordinary differential equations for the differential equation (1) with matrices M and K whose entries are defined as for k = 1, ..., N and l = 1, ..., N . Applying the spectral method to the static beam equation (6) yields the initial vector b(0) of the differential equation (8) through the linear equation Here, the vector v is given by the entries v k = P ϕ k (L) (11) Fig. 1: Deflection at the ski shovel. Fig. 2: Single-sided amplitude spectrum of the deflection.
for k = 1, ..., N . The initial velocity vector b t (0) of the differential equation (8) is set to zero. The system of linear ODEs (8) is transformed to a first order system of linear ODEs, which is solved by the matrix exponential function. Finally, the deflection at the ski shovel is determined by the Ansatz (7) and their eigenfrequencies are analysed by the fast Fourier transform (FFT).
3 Results Fig. 1 shows the deflection at the ski shovel. The single-sided amplitude spectrum of the deflection at the ski shovel is depicted in Fig. 2. The most dominant eigenfrequency is about 13 Hz.

Discussion and Outlook
In this study we proposed an approach to study the vibrations of an alpine ski, taking into account material variability. In a next step, the random field r(x), which represents the variation in the bending stiffness of the ski, will be generated based on lab measurements to validate the approach. In addition, the presented approach will be implemented in a simulation model of a multi-body skier [5] to study the ski vibrations during a simulated run. A sequence of random runs will be performed (Monte Carlo simulation) and the vibrations will be analysed. In the future, the presented approach will be applied to other structural problems, as for example the vibrations of a composite bridge under traffic load.