bobsleigh optimization

ENOC 2011, 24-29 July 2011, Rome, Italy

Bobsleigh Optimization - A Customized Dynamic Vibration Absorber with Limit Stops

Pascal Arnold∗, Christoph Glocker∗∗Institute for Mechanical Systems, Department of Mechanical and Process Engineering,

ETH Zurich, Zurich, Switzerland

Summary. This paper presents a dynamic vibration absorber for a bobsleigh, that is based on vibration measuring runs on a real track.Observed chassis resonances are reduced by means of an absorber with the assumption that smaller vibration amplitudes should enhancethe performance and controllability of the bobsleigh. The device features unconventional elements such as dry friction damping or limitstops for the tuned mass.

Bobsleighs and Vibrations

The performance optimization of bobsleighs is a challenge that engages a variety of research fields. Whereas aerodynamicoptimization and athletic training are advancing in almost every competing team, the merit of vibration analysis of bob-sleighs is a topic of much debate. To gain more information about the nature of the vibrations occurring in a bobsleigh,track experiments with a bobsleigh depicted in figure 1 have been conducted using a variety of sensors. In this paper thesensors around the steering headset are of interest.

1

3

2

Figure 1: Raw bobsleigh chassis with the canopy removed; accelerometer at sliders (1), strain gauge to measure the slider load (2) andaccelerometer at the steering headset (3).

Figure 2 shows the logged slider acceleration in the upwards direction of the bobsleigh. After smoothing the signal thatfeatures rather large variation (b), one can see the centrifugal accelerations (a) of the bobsleigh in corners. When the bob isdriving through corners, the upwards acceleration direction is inclined to the gravity direction. At times t = [62 . . . 64.5]s,the bob is on a straight part of the track with a little bump around 63.3s. In the time window between 65s and 67.5s thebob passes a corner.

58 59 60 61 62 63 64 65 66 67 68

-10

0

10

20

Time t [s]

Sli

der

Acc

eler

atio

n[m

/s ]2

Measured

Smoothed

a

b

Figure 2: Measured slider acceleration on the Cesana bobsleigh track. Centrifugal acceleration (a) is of the same magnitude than thevariation of the signal (b).

Standard FFT analyses of several similar signals lead to unsatisfactory results, as it is impossible to find a balance betweenanalyzed signal length, windowing and FFT length, such that the FFT variance is acceptable. As the acceleration datahas been sampled at 2kHz and the data set size is approximately 180’000 points or 60s, it is expected that a reasonablefrequency resolution and FFT precision could be achieved between 5 and 100Hz. This means that the signal sampling isappropriate.


Another measure for the frequency domain properties is the Power Spectral Density (PSD) derived after Welch’s Method[4]. In this analysis, the signal is chopped into several (50% overlapping) segments which are individually subjectedto a FFT analysis. Each segment is used to compute its modified periodogram after [4], and the final PSD measure isan averaged value of these periodograms with units Power per Hz. This power spectrum estimate features a reducedvariance, as well as reduced signal amplitude loss due to the windowing process compared to a single FFT analysis.Therefore a signal with high variance is much more convenient to interpret. The drawback of this method is a lowerfrequency resolution because of shorter individual FFT segments.To compute the PSD for the bobsleigh measurements, the signal is chopped into 88 segments (4096 points each or approx.60m track length) with 50% overlap, weighed by a Hamming window. As a consequence, one can clearly distinguish theresonance peaks from noise, despite the rather low frequency resolution of 0.48Hz. Figure 3 depicts the PSD of threesignals logged by accelerometers and strain gauges in the bobsleigh during a complete track run. The power spectrumof the upwards slider acceleration makes clear, that the hard contact between slider and rough ice surface induces prettymuch a wide-band excitation into the bobsleigh chassis with the expected natural decreasing signal energy for higherfrequencies.

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0

10

20

30

Frequency f [Hz]

PS

D[d

B/H

z]

Slider Acceleration

Slider Load

Steering Headset Acceleration

Frequency Resolution 0.48Hz

Figure 3: Power Spectral Density (PSD) of three measured signals during a complete bobsleigh run.

It is also found that a characteristic resonant phenomenon occurs in the complete front substructure of the chassis. Startingat the upwards slider load sensor (see figure 1, (2)), the vibration is transferred to the front axle and the upwards headsetacceleration in figure 1, (3). From figure 2 it becomes clear that the vibrations are of the same order of magnitude than thecentrifugal acceleration, thus a≈b. Based on the assumption that large vibrations in the bobsleigh chassis might have anegative impact on controllability or performance, a Dynamic Vibration Absorber (DVA) system is developed to reducethe vibration energy of the upwards steering headset acceleration.

Identification of the Dynamic Vibration Absorber Target Frequency

In order to design a DVA featuring optimal efficiency, it is crucial to assign a relevant resonance peak that is presenton any bobsleigh track. Figure 3 shows two predominant peaks at 16 and 25Hz that could be observed in several tests.Laboratory measurements using hammer strokes at different spots for excitation reveal that the well-reproducable 25Hz-peak is related to the first bending mode of the chassis. It covers less than 3% of the signal energy contained between 5and 100Hz (application of Plancherel’s theorem [5]). On the other hand, the 16Hz-peak is clearly a consequence of theinteraction between the bobsleigh and the track, as it is absent in all laboratory tests, but present on the field tests. Thispeak contains substantial 13% of the signal energy. Several test runs on different bobsleigh tracks have been performedand the named peak frequency has always been found in the range between 14 and 16.5Hz. Possible reasons for variationsare different ice conditions during the test sessions or different track foundation properties (natural track vs. artificiallycooled track). The DVA target frequency is set to 16Hz mainly due to the higher energy absorbing potential. Even if theresonance peak is not perfectly matched for all tracks, the DVA absorbs more energy than if it was tuned for the 25Hzpeak.


DVA with Limit Stops and Dry Friction Damping

FN

m1

m2

k1

k2

d1x1

x2

xmax

xmax

r t( )

¹

Figure 4: Dynamic vibration absorber withlimit stops (xmax) & dry friction damping (µ).

To determine appropriate parameters of the DVA a simplified model of thebobsleigh front substructure is required (see figure 4). As vibrations haveonly been analyzed in the upwards direction, the model should also be one-directional. The mass of the surrounding bobsleigh structure is condensedin m1 = 31kg and parameters [k1 = 3.25 · 105N/m, d1 = 1.6 · 103Ns/m]are chosen such that the 16Hz peak can be reached and the damping is cor-responding to the measured PSD in figure 3.In contrast to many engineering applications, viscous dampers are forbiddenby the bobsleigh federation rules [3]. As a consequence, DVA damping isrealized by dry friction (µ = 0.3, FN ) between the moving mass m2 and thefront structure m1. Also the spring k2 cannot be implemented as a simplecoil spring, but has to be designed as a leaf spring in order to be consistentwith the rules. The measured acceleration data (e.g. figure 2) of the sliders isused to generate the base excitation r(t).An initial analysis and optimization of this simplified model leads to DVAparameters showing relatively high amplitudes (x2 − x1) between the tunedmass and the underlying body, which would result in interference ofm2 withthe surrounding structures of the bobsleigh. The DVA system is therefore ex-tended by two limit stops (xmax = 0.01m in figure 4) to bound the amplitude of m2. The equations of motion

m1x1 = k1(r − x1) + d1(r − x1) + k2(x2 − x1) + λN1 − λN2 − λTm2x2 = k2(x1 − x2)− λN1

+ λN2+ λT

(1)

of this specific multibody system with impacts at the stops (λNi) and dry friction (λT ) are formulated within the nons-

mooth dynamics aproach [1]. The set-valued force laws of normal cone type

−λN1∈ Upr(x1 − x2 + xmax)

−λN2 ∈ Upr(x2 − x1 + xmax)−λT ∈ µFN · Sgn(x2 − x1)

(2)

are formulated as inclusions using Upr and Sgn functions (figure 5). The unilateral contacts induced by the limit stops(eq. (2)) are considered to be hard constraints with a Newton-type of impact law with a corresponding impact coefficientεN = 0.2. For the tangential contact, εT = 0 is chosen. Moreau’s timestepping algorithm, which is a time-discretizationon velocity impulse level, is used for numerical simulation [2].

gTgN

Upr( )gNSgn( )gN

1

-1

Figure 5: Upr and Sgn-functions used in the inclusions of equation (2).

For the further analysis the one-dimensional model with the DVA attached as seen in figure 4 is named the 2-DOF systemand the reduced model only considering m1, k1, d1 and r(t) without DVA attached is defined as the 1-DOF system.


Numerical Optimization of the DVA

The main purpose of the DVA is to reduce vertical vibrations of the steering headset. Because there exist no optimal designrules for a system with dry friction and limit stops, a cost function J is chosen to be minimized, taking the accelerationroot mean square value

J =√∑

(x1)2/N

of the underlying massm1. N is the number of simulation time steps. A close connection between the cost function J andthe power spectrum is given, because the integral over all frequencies f of the power spectral density is equal to J2. Analternative cost function such as the amount of absorbed energy of the DVA results in similar values for the optimizationparameters. Figure 6 shows that the 1-DOF model approximates the measured 16Hz peak adequate, and that the 2-DOFsystem shows absorbing properties in the frequency band between 15 and 22.5Hz (see detail in figure 6).

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0

20

40

Frequency f [Hz]

PS

D[d

B/H

z]

Frequency Resolution 0.48 Hz

2

1

23

Measured

Simulated 1-DOF Model

Simulated 2-DOF Model

1

3

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0

10

20

Absorbing Region

Detail of the 16Hz Peak

Frequency f [Hz]

1

Frequency f [Hz]

2

3

Figure 6: PSD signals of measured upwards steering headset acceleration and simulation results for the 1-DOF and 2-DOF model.

To quantify the absorption effect of the DVA, one can estimate the reduction of vibration energy by a comparison of thesignal energies of the 2-DOF system with the 1-DOF system. The corresponding total signal energies are equal to theintegral of the PSD functions, calculated in figure 6, over all frequencies. However after studying figure 6 it becomesclear that the 1-DOF model is fitting the measured data only in a small frequency band ftrust = [5 . . . 22]Hz accuratelyenough. In this region of trust, only about 23% of the total signal energy of the measurements are contained. Anyhow, thefrequency interval of trust is always similar, no matter what measurements the 1DOF-model is based on. Because thereonly exist PSD values for discrete frequency intervals (0.48Hz), the signal energy is a sum and therefore the vibrationenergy reduction

ρDVA = 1−

22∑f=5

PSD2DOF (f)

22∑f=5

PSD1DOF (f)

∈ [0 . . . 1].

In order to verify the optimal parameter choice, the same DVA setup is tested for other data on 3 different tracks. Con-sidering the frequency band ftrust = [5 . . . 22]Hz, the signal energy can be reduced by ρDVA = 15.6% on the Cesanatrack, ρDVA = 16.8% on the Igls track and ρDVA = 7.2% on the St. Moritz track when a DVA is fitted to the system.It is found, that the performance of the DVA is relatively insensitive to parameter uncertainties, and that the version withlimit stops seems to damp away also frequencies that are lying outside of the working frequency band of the DVA withoutthe stops - mainly due to the high energy absorption of the impacts. Because on the natural ice track in St. Moritz a lessdominant 16Hz-peak is observed, the corresponding signal energy reduction ρDVA is obviously smaller.


With optimal parameters, the movement of m2 can feature both sticking at the dry friction contact and bumping into thelimit stops (xmax), as can be seen in figure 7. It shows a simulation of the system with the optimized DVA attachedas described in figure 4, excited by the measured slider acceleration r(t). Optimal performance was found with k2 =5 · 105N/m and µFN = 71.5N.

224.6 224.8 225 225.2 225.4 225.6 225.8 226 226.2 226.4

-0.01

-0.005

0

0.005

0.01

Time t [s]

x

-2

x1

[m]

StickImpact

Figure 7: Simulated time signal of the relative motion (x2−x1) of the DVA on a bobsleigh track. Sticking at x2−x1 = const., impactsat x2 − x1 = ±0.01m.

Realization and Testing

The promising and robust simulation results of the optimized DVA led to its production, that, thanks to the limit stops forthe motion of m2, can be realized in a relatively small package. Up to now, only four test runs with a 4-man bobsleighon the tracks of Winterberg (Germany) and St. Moritz (Switzerland) have been conducted. The DVA has a noticeableinfluence at high speeds and stronger vibrations, according to the testpilot’s observations, but the number of test runs isby far too small to give well-founded statements about the performance of the device. Even if the simulation showed alarge parameter tolerance of the model, it remains an open task to verify the vibration reduction effect by accelerationmeasurements onm1 andm2 or even direct displacement measurements (x2−x1) during a bobsleigh run. Also a completetest campaign to evaluate the DVA performance on the track with 2-man bobsleighs is planned.

Conclusions

The analysis of measured vibration signals during a bobsleigh run reveals resonance phenomena in the steering headset.Due to the high variance of the data the Power Spectral Density is applied for interpretation of the signal frequency con-tents. A dynamic vibration absorber with dry friction damping and limit stops has been designed and optimized for energyabsorption in a bobsleigh chassis. Limit stops to bound the tuned mass movement have been implemented mainly due tomechanical constraints, but they also enhance the working bandwith and parameter tolerance of the device. Simulationsusing a one-dimensional model predict that the vertical steering headset vibration energy is reduced by 7.2-16.8% whenthe absorber is attached, depending on the track that is simulated. Not enough experiments have yet been conducted toverify the effect of the device on race performance.

References

[1] Glocker Ch. (2001) Set-Valued Force Laws: Dynamics of Non-Smooth Systems. Lecture Notes in Applied Mechanics 1, Springer. Berlin Heidel-berg. 222 pages.

[2] Leine R.I., Nijmeijer H. (2004) Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lecture Notes in Applied and ComputationalMechanics Vol. 18. Springer. Berlin Heidelberg New York.

[3] International Rules Bobsleigh, Federation Internationale de Bobsleigh et de Tobogganing (FIBT)http://www.fibt.com/fileadmin/Rules/Rules%202010-2011/Reg.BOB-2010-E.pdf, p. 29-49.

[4] Welch P.D. (1967) The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short,Modified Periodograms. IEEE Trans. Audio Electroacoustics AU-15: 70-73.

[5] Wiener N. (1988) The Fourier Integral and Certain of its Applications. Cambridge University Press.

bobsleigh optimization

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