Churyumov-Gerasimenko

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A method for Inverting the touchdown shock of the Philae lander on comet 67P/Churyumov-Gerasimenko Claudia Faber, Martin Knapmeyer, Reinhard Roll, Bernd Chares, Silvio Schröder, Lars Witte, Klaus Seidensticker, Hans-Herbert Fischer, Diedrich Möhlmann, Walter Arnold www.elsevier.com/locate/pss

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S0032-0633(14)00382-1 http://dx.doi.org/10.1016/j.pss.2014.11.023 PSS3862

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Planetary and Space Science

Received date: 4 September 2014 Revised date: 6 November 2014 Accepted date: 19 November 2014 Cite this article as: Claudia Faber, Martin Knapmeyer, Reinhard Roll, Bernd Chares, Silvio Schröder, Lars Witte, Klaus Seidensticker, Hans-Herbert Fischer, Diedrich Möhlmann, Walter Arnold, A method for Inverting the touchdown shock of the Philae lander on comet 67P/Churyumov-Gerasimenko, Planetary and Space Science, http://dx.doi.org/10.1016/j.pss.2014.11.023 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

A Method for Inverting the Touchdown Shock of the Philae Lander on Comet 67P/Churyumov-Gerasimenko Claudia Faber1, Martin Knapmeyer1, Reinhard Roll2, Bernd Chares2, Silvio Schröder3, Lars Witte3, Klaus Seidensticker1, Hans-Herbert Fischer4, Diedrich Möhlmann1, and Walter Arnold5 1

DLR Institute of Planetary Research, Berlin, Germany Max Planck Institute for Solar System Research, Göttingen, Germany 3 DLR Institute of Space Systems, Bremen, Germany 4 DLR MUSC, Cologne, Germany 5 Saarland University, Department of Materials Science and Technology, Campus D 2.2, and 1. Physikalisches Institut, Georg-August University, Göttingen, Germany 2

Abstract The landing of Philae on comet 67P/Churyumov-Gerasimenko is scheduled for November 12th, 2014. Each of the three landing feet houses in one sole a tri-axial acceleration sensor of the Comet Acoustic Surface Sounding Experiment (CASSE). They will thus be the first sensors to be in mechanical contact with the comet surface. CASSE will be in listening mode to record the deceleration of the lander when it impacts with the comet at a velocity of approx. 1 m/s. The signals are used to discern from these data whether the selected landing place is of icy or sandy nature. Here, a deeper analysis of this data yields information on the reduced elastic/plastic modulus of the comet’s surface material. Keywords: Comet 67P, Surface Elastic-Plastic Properties, Landing Shock, Rosetta-Philae

Corresponding author: W. Arnold, Current address: Department of Material Science and Materials Technology, Saarland University, D-66123 Saarbrücken; Email: [email protected]

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Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

I.

Introduction

In August 2014 the European Space Agency’s spacecraft Rosetta encountered the short-period (orbital period 6.45 yr) comet 67P/Churyumov-Gerasimenko. Rosetta carries the lander spacecraft Philae (Biele and Ulamec, 2008; Schulz et al., 2009) that will attempt to land on the comet’s nucleus in November 2014. It is expected that both the Rosetta orbiter and the lander craft will be able to study the nucleus and its environment in detail as the comet journeys through the inner Solar System, while the comet activity varies as the icy surface increases its temperature. In this paper we discuss whether it is feasible to discern information on the mechanical properties of the comet surface by monitoring the landing shock. To this end, landing tests were carried out at the Landing & Mobility Test Facility (LAMA) of the DLR Institute of Space Systems in Bremen. These data are presented and their inversion is discussed. It is expected that the same procedure can be applied to the deceleration signal measured when landing on comet 67P/Churyumov-Gerasimenko. The derived analytical expressions allow a rapid estimate of the comet surface elastic and plastic properties. A detailed modeling of the comet soil mechanical properties may follow together with sound velocity data obtained by the Comet Acoustic Surface Sounding Experiment (CASSE) (Kochan et al., 2000; Seidensticker et al., 2007) and data from the Philae thermal and mechanical properties probe MUPUS (Spohn et al., 2009) II.

Landing Test Experiments

At the LAMA facility, a six-axis robot KR500 manufactured by Kuka/Germany with a nominal bearing capacity of 500 kg sits atop a rail track having a lateral travel distance of 10 m. It can be programmed to move landers or rovers along predefined paths (Witte et al., 2014) to study landing procedures with predefined velocities ranging from vx = 0.1 to 1.1 m/s, see Fig. 1. For the landing tests, the lander model had the same mass, center of gravity and moments of inertia, and outer measures like the flight model Philae. The attached landing gear was the qualification model. It consists of an original landing gear of three legs manufactured of carbon fiber and metal joints (Biele et al., 2009). Its damping mechanism was identical to the flight model. A dead mass of the size and mass of the lander housing is attached via the damper above the landing gear to represent the lander structure as a whole. The damping system is an electromechanical device which provides an

Fig. 1. LAMA test facility: The lander Philae is suspended at the hand of a robot. Between the stiff hand and Philae, there is a spring kLT = 4.1 kN/m. It serves to compensate for gravity by a programmed upward movement of the hand as soon as the lander touches down on the concrete floor and on sand kept in large buckets.

equivalent viscous force (Witte et al., 2013) with a damping constant of γD = 601 Ns/m. Attached to each leg is a foot with two soles and a mechanically driven fixation screw (“ice screw”) to secure the 2

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

lander on the comet. The lander model used in our experiments was not equipped with the two harpoons used on Philae to fix it to the ground (Komle et al., 1997). The right soles, if viewed from the outside towards the lander body, house a Brüel & Kjaer DeltaTron 4506 triaxial piezoelectric accelerometer as used on the spacecraft whereas the left sole houses the CASSE Orientation of the three axes was such that one of the axes, here the x-axis of the accelerometer, points downwards, while the y and z-axes are horizontal. Therefore, we designate the landing velocity as vx, although this is not in line with the usual designation of the velocity vector perpendicular to a surface. Data were recorded at a sampling rate of 8.2 kHz within a time gate of 2 s. In parallel a video sequence was taken, in order to monitor the touch-down on concrete and sand together with the movement of the fixation (ice) screws. III.

Materials and their Characterization

Touchdown measurements were conducted with landing velocities between 0.1 to 1.1 m/s on three types of grounds: the concrete floor of the LAMA facility, and two different types of sands. In order to protect the lander from damage, landing velocities on concrete were limited to 0.1 and 0.2 m/s. The two sands were kept in 0.4 m high buckets having a volume of 90 liters (Fig. 1). One was fine-grained quartz sand, and the other sand an equivalent for Mars soil (MSS-D). Its SiO2 content was 99.6%, with a mean grain size of 0.22 mm and specific surface area of 111 g/cm2. The remaining aggregate material was Al2O3 (0.2%) and Fe2O3 (0.04%), and some other materials. MSS-D was produced by the DLR. Its content is again quartz sand (called H33) + olivine in a mixing ratio of 50 wt% each as specified in an internal product sheet. The particle size distribution of the MSS-D sand peaked at 0.2 mm, with a wide distribution extending from 0.06 mm to 0.4 mm. Both WF 34 and H33 were obtained from the group of the Quarz Werke, Haltern, Germany, see http://www.quarzwerke.com/de/. IV.

Signal Shapes and Number of Data

Examples for deceleration data are shown in Fig. 2a. Only the first impulse of 20 ms length, obtained with each foot of the landing gear, was evaluated with the procedures described in this paper. As can be seen, this first deceleration signal and hence the force varies sinusoidally for a half period, see Figs. 2a and 2b. The frequency of the signal was estimated from a least-square fit of the halfperiod of length Tc after a first visual check of the signal. Signals from the three feet were usually well separated in time. In altogether 28 landing tests, in only about five cases the deceleration signals overlapped because two or three feet touched down simultaneously. Because the landing gear has three

(a) (b) Fig. 2. (a) Signals obtained from a touch-down test on the sand MSS-D at a velocity vx = 0.1 m/s (measurement # 9). The vertical deceleration (indicated by negative values) component for all three feet is shown. Note that for the landing velocities vx < 0.8 m/s the fixation screws did not touch the surface of the sand during the first deceleration signal analyzed, and hence the first signal was only determined by the restoring force of the sole-surface contact; (b) Dashed-line in red: Sinusoidal fit of the mean deceleration impulse measured with a landing velocity of vx = 0.1 m/s on MSS-D. The half period can be clearly measured from the contact time, here Tc/2 = 6.6 ms (measurement #9, foot 1, see (a)). Arrow: Contact-resonances of the soles having a frequency of fcr ≈ 606 Hz are superimposed on the deceleration pulse (see section VII). They are also visible in the signals of foot #2 and #3 in (a) if the time scale is expanded as well as in the Fourier transforms of all three signals.

legs each with a foot each containing one accelerometer, the number of possible data sets for the acceleration data in x-direction is 28×3 = 84 (data in y- and z-direction exists as well but are not reported here). From these, seventy data sets could be analyzed unambiguously. No standard deviations are 3

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

given for the data because each test is an individual measurement with slightly different conditions both in the trajectory of the touch-down as well as in the condition of the sand in the buckets. Some signals show a positive overshoot. From the videos taken, one can see that in these cases the sole housing the CASSE transmitter touches down first giving the sole housing the accelerometer a slight initial down-ward push and hence a positive acceleration. The oscillations of the lander structure display a quite rich frequency spectrum and we cannot assign presently all observed frequencies to the oscillations of individual components of the lander structures. Finite element modeling will help in this task at a later stage. V.

Lander Mechanical Model

In order to analyze the deceleration data in a simplified way, it is proposed to view the lander structure as a mass-spring-damper system representing a mechanical series circuit coupled to the soil by a contact spring, whose stiffness is determined by the reduced elastic moduli of the soil-sole contact and the contact radius (Fig. 3). At the end-point A of the spring kLG, the force K(t) is generated by the restoring forces in the contact either in the LAMA tests or when landing Philae on comet 67P/Churyumov-Gerasimenko. The landing shock causes a deceleration b(t) which is registered by the accelerometer firmly attached in one sole of each of the three foot-sole pairs. For sinusoidal forces or decelerations, an analytical inversion is possible. By approximating the velocity as a periodic function v(t) = v0exp(jωt) during the phase of elastic contact, we can relate the acceleration to the velocity via v(t) = v0ωexp(jωt+π/2). In the equivalent electrical circuit, the velocity corresponds to the current injected (Fig. 4). It is straightforward to calculate the mechanical admittance Yms = v(t)/K = b(t)/(jωK) in analogy to the admittance Yep of the electrical parallel circuit where K is the force acting on the soles. With the compliance F = 1/(kLG + kLT) this leads to

(

2 % = K 1/m L + jω/ γ D - ω /(k LG +k LT ) b(ω)

)

(1)

% where b(ω) is a complex quantity. Here, the compliance of the springs 1/(kLG + kLT) is equivalent to the capacitor C, the mass mL to the inductance L, and the damping γ corresponds to the resistance R in the analog electrical parallel circuit shown in Fig. 4. Note that the two springs kLG and kLT are effectively operating in parallel, because one is extended (kLT), whereas the other is compressed (kLG) by the lander mass mL. The videos taken show that the reaction of the robot hand to compensate for the gravity force sets in after a time delay larger than 20 ms, larger than the impulse durations considered here.

Fig. 3. Mechanical model for the lander structure. Parameters are given in Table 1.

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Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

Table I: Measured parameters of the lander model (Witte et al., 2014) Parameter Symbol Stiffness of a single leg of kLG the landing gear Mass of the lander mL Damping constant on the γD lander structure Stiffness of the suspension kLT of the lander structure

Value 13.3 kN/m 98 kg 601 Ns/m 4.1 kN/m

Fig. 4. Mechanical equivalent model as a series circuit in analogy to an electrical parallel circuit based, for periodic loading at a frequency ω/2π. At the end-point A (Fig. 3) of the spring kLG, the force K(t) is generated by the landing shock, causing a deceleration b(t) which is registered by the accelerometer mounted in one of the three feet in each leg. The mechanical admittance Yms of the series circuit corresponds to the electrical parallel admittance Yep (Meyer and Guicking, 1974).

We may define a relation for the maximal force K0 between the sole and surface material, so that

K 0 = E* × A e

( 2)

where Ae is an effective contact area and E* (definition see below) is the reduced modulus of the solesurface material contact. Calculating the absolute value of b(ω) = Re b% + Im b% of Eq. (1) and using Eq. (2) yields 2

2  1  ω2 ω b max = K   + 0  m L k LG +k LT  γ 2   D

with

ω

0

=

= K

0

2 1  ω 1 2 2 mL  ω0

2

2  ω  +  2 γD 

( k LG +k LT ) / m L

(3)

(4)

Inserting the value from Table 1 yields ω0 = 13.3 s-1 and f0 = 2.12 Hz. The damped eigenfrequency for the mechanical series circuit of Fig. 4 is given by

ω

d

=

 k +k  2 ω0 −  LG LT   2γ D 

2 = ω 1 − 0

( k LG + k LT )m L 4γ D

2

= ω 1− D 0

2

(5)

where D is the normalized damping constant D = ω0mL/2γD. Inserting the values of Table I yields D ≈ 1.08, i.e. the landing gear of Philae represents a slightly overdamped oscillator and hence no oscillations of a landing arm should be observed upon landing impact. However, typically two oscillations of a single leg of the landing gear upon impact with a few mm amplitude can be seen in the high-speed videos recorded during the LAMA tests, and their frequency is fd ≈ 1.7 Hz, approximately the same for landings on concrete and on sand. One should keep in mind, that the damping mechanism is an electro-mechanical device which develops its full effective viscous damping value γD = 601 Ns/m only after a few ms. There are moving masses within the lander structures which causes these oscillations 5

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

and hence are not the damped eigenfrequencies of the structure. We do not want to discuss the phenomena further because it does not influence the analysis of the data which follows. In all tests carried out ω = π/Tc was larger than 150 s-1 up to 2000 s-1, hence ω >> ω0. Given the parameters in Table 1, it is easy to see that for all measured ω, Eq. (3) can be simplified to:

b max = K

ω 0

2

2 m L ω0

(6)

which is independent on the damping constant γD of the landing gear. On the right-hand side of Eq. (6) are the quantities ω = π/Tc, ω0, b max which are measurable, and mL is known. This allows one to derive the maximal force K0 acting on the sole and caused by the contact-forces. As can be seen K0 is determined by the restoring force in the contact sole-calibration material, here concrete, MSSD, and WF 34. The forces in the contact are mitigated by the transfer function of the lander structure. For ω >> ω0 this function converges to (ω/ω0)2 entailing that the acceleration acting on a given sole is reduced by the compliance of the lander structure. In Eq. (2), E* may be purely elastic or of plastic nature, or containing both elements. We will discuss this in the course of the paper. VI.

Forces Philae Sole-Surfaces

Elastic Restoring Forces on Concrete If the restoring forces were purely elastic, there is no permanent or plastic strain stored in the contacting materials, i.e. the soles and the test surface or the comet surface. In this case, the lander structure would re-bounce gaining the original velocity in opposite direction, here –vx. Of course there is the damping mechanism in the lander structure, there is plastic deformation in the surface material, and there are the actions of the fixation screws in each sole pair, the action of the harpoons, and additional forces on a comet (Komle et al., 1997). Without going into details, the harpoons are fired after a delay of at least 20 ms after two feet have touched the surface of 67P, and hence after the signals we discuss here. A purely elastic contact is a hypothetical case, yet it is important to understand its ramifications. In Hertzian contact mechanics, the force K caused by an elastic impact is given by (Johnson, 1985): K=

4 1/2 * 3/2 3/2 R E δx = K ' δx 3

(7)

Here, E* is the reduced elastic modulus of the contacting materials 1/E* = (1-υ12)/E1 + (1-υ22)/E2

(8)

where the υi are the Poisson ratios of the contacting materials, and 1/R = 1/R1 + 1/R2 is the reduced radius of the sole-surface contact. If R2 >> R1, R is the sole radius. The deformation depth of the two colliding partners is designated δx. In an elastic Hertzian contact, δx is defined as the distance of two points of the colliding partner far away from the contact. Always both contacting materials get deformed. The contact radius a is given by

a = Rδ x

(9)

The maximal force is given by K0 =

1/2 4 1/2 * 3/2 4 2 *  * R E δ x,0 = E δ x,0 = k δ x,0  Rδ x,0  3 3 3 

(10)

Here, k* = 2aE* is the contact stiffness. The deceleration data as a function of time is shown in Fig. 5. Despite the fact that the restoring force versus distance is non-linear, the penetration curve can be approximated by a sinusoidal function ((Johnson, 1985). Inserting Eq. (9) into Eq. (10), one obtains the following relation between the maximal force K0 and the measured deceleration 6

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

K = 0

2 ω0 m L ω

2

b max =

3 * 4a E 3R

(11)

The expression 4a3/3R is the effective area Ae in Eq. (2).

Fig. 5. Theoretically expected relative elastic collinear force K(t)/K0 acting on two contacting spheres of reduced radius R and the corresponding relative penetration depth δx/δx,0 versus the relative time scale t/t* (2t* = Tc). The dashed line represents a sine function (Johnson, 1985).

Fig. 6. (a) Deceleration versus time for the landing test on concrete (measurement # 5, foot 2, vx = 0.2 m/s); contact-resonance frequency fcr = 2.5 kHz; (b) Deceleration versus time for the landing test on concrete (measurement # 3, foot 2, vx = 0.1 m/s); fcr = 2.3 kHz. In contrast to the landing tests on sand, see Fig. 2b, the contact resonances frequencies are higher and more pronounced. Yet, the overall sinusoidal behavior of the first contact impulse can still be observed with a contact time of Tc = 2.6 ms in (a) and in (b) Tc = 2.3 ms. Note that there is frequency dispersion in the contact-resonance frequency due to the time-varying dynamic load of the landing shock.

No permanent depressions remained in the case of the landing tests on concrete, neither in the concrete, nor in the soles, and the envelope of the signals was approximately sinusoidal, see Fig. 6. However, additional signals from the contact-resonances were very well visible, see discussion in section VII. For the data shown in Fig. 6a for concrete (vx = 0.2 m/s, bmax ≈ 1300 m/s2, Tc ≈ 2.5 ms), the effective angular frequency is ω = π/Tc = 1.26×103 s-1. With ω0 = 13.3 s-1, Eq. (6) yields K0 ≈ 14.3 N. Furthermore, we can estimate the displacement of the accelerometer to be u0 = bmax/ω2 ≈ 0.82×10-3 m. If the accelerometer was firmly attached to the sole, u0 ≈ δx,sole should be valid. This entails a = (δx,sole ×R)0.5 ≈ 12.8 mm. From Eq. (10) we derive δ x,0 =

 1   1  3 3 1 3 1 1 1 1  = K  K = K  + + + + 2 0 k∗ 2 0  k cont ksole (k LG + k LT )  2 0  2aE∗ ksole (k LG + k LT ) 

(12)

Here, kcont is the stiffness of the sole-concrete contact, kLG + kLT = 17.4 kN/m, and ksole = 173 kN/m (see Appendix), and Ee* is the reduced modulus of a concrete-glass fiber epoxy contact which is about Ee* ≈ 12 GPa, because the reduced E-modulus of the aggregrate material concrete is Ec* ≈ 30 GPa 7

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

(Ashby et al., 2009; Hashin and Monteiro, 2002) and the reduced E-modulus of the glass-fiber composite is Eg* ≈ 20 GPa (Schwartz, 1992). Therefore kcont = 2aEe* ≈ 307 MN/m. It is easy to see from Eq. (12) that the deformation of the lander structure on concrete or other materials having a high elastic modulus, is only determined by its compliance (kLG + kLT)-1 and not by the compliance of the contact sole-concrete, 1/kcont . Whereas one can derive from such measurements the stiffness of the landing gear, one cannot calculate the elastic modulus of materials like concrete, rock or ice (Eice* ≈ 9 GPa (Hobbs, 1974); in this case kcont ≈ 230 MN/m) from the envelope of the landing shock. This is because when landing on such materials, the deceleration is determined by the compliance of the landing gear of Philae, and then the data cannot be reliably inverted. The Philae landing gear behaves as designed, namely to reduce the forces acting on its structure during the landing shock. If we set 2aE* = kLG (for the flight model kLT = 0) we obtain E* = 0.27 MPa for a contact radius of a = 2.5 cm. It will be difficult to measure reduced moduli much larger than this value by analyzing the time-evolution of the landing shock. Restoring Forces on Sand When the soles in the LAMA test touches down on sand, in all tests a permanent depression remained and hence plastic deformation or strain was stored. When loading a granular material, there are first elastic restoring forces at small loads and then plastic deformation sets in, i.e. the sand consolidates. This process is rather complex and is described in many textbooks of Geotechnology and Civil Engineering (Kolymbas, 2011; Lang et al., 2011). The corresponding tests are also called odometer tests, if the sidewise movement of the soil is suppressed by placing the sand in a container. The compression modulus ME in these tests is defined such that MEO = dσ/dε, where σ is the stress exerted on the mobile top plate of the container of height h0 and ε is the strain in the container, i.e. ε = ∆h/h0 of the sand pile. The modulus ME depends on the absolute strain values and shows hysteresis when the soil is loaded again after unloading (Kolymbas, 2011; Lang et al., 2011). There is also the Bevameter test (Wong, 1980). Here, one presses a plate into the soil which is not confined, and one measures the load per area σ versus the sinkage h of the plate into the soil. The corresponding modulus is often called deformation modulus MEp = (dσ/dh)×d, where d is the diameter of the plate. Sometimes the multiplication by d is omitted and the modulus has the unit N/m3. It is then called the bedding modulus MB. It is a measure of the bearing capacity of soils. Measurement procedures are standardized (e.g. DIN 18134) and there are empirical relations to the shear strength of the tested sand which depend on the type of soil, its porosity, water content, and friction angle. However, there is no generally valid relation. There is an ongoing debate to find such physically based relations. These moduli should not be confused with the elastic modulus E, because they are parameters characterizing plastic deformation. In order to obtain more information, Bevameter tests were carried out by pressing a complete foot (Fig. 7a) into MSS-D and WF 34 with the (i) screw fixed and (ii) with the screw operational like in the actual FM module of Philae. Forces were measured in the robot hand. The sole pair was pressed with constant velocity of 1 mm/s into the sand for 15 mm and for 50 mm when the fixation screw was operational. The force was recorded as a function of time and hence the distance into the sand (Figs. 8a and 8b) and the geometrical parameters of the ensuing depression were measured as well (Figs. 7b and 7c). Evaluation of the Data on Sand For the LAMA landing tests on both sands K0 was first calculated for all measurements for veloci-

ties 0.1 ≤ v x ≤ 0.8 m/s. From the first deceleration signal versus time, see Figs. 2a and 2b, ω was determined like in the case for concrete. Together with ω0 = 13.3 s-1, the maximal K0 was determined according to Eq. (6). Again, for all measurements ω >> ω0 held, and the fixation screws were not actuated. For the quartz-sand olivine mixture MSS-D, one obtains 28 N > K0 > 12 N, and averaged over 20 measurements K0,av = 18.6 N. For the quartz-sand WF 34, one obtains 16.3 N > K0 > 3 N and averaged over 18 measurements K0,av = 8.3 N. Using the geometrical relations for a spherical segment, it is easy to calculate the penetration δx with a sole radius R = 200 mm and an estimated contact radius a = 2.5 ±0.5 cm taken from the pictures during the LAMA test. One obtains δx ≈ 1.6 ± 0.8 mm for MSS-D. Using Si ≈ 12.5 N/mm from Fig. 8a, the average measured K0 = 18.6 N corresponds to a penetration of δx ≈ 1.5 mm and a calculated contact-radius of a = 2.4 cm, very close to the value expected. 8

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

(a) (b) (c) Fig. 7. (a) Calibration experiments were performed by pressing a complete foot into MSS-D and WF 34 with the (i) screw fixed and (ii) with the screw operational like in the actual FM module of Philae. The sole pair was pressed with constant velocity of 1 mm/s by the robot-handling system used in the LAMA test into the sand for 15 mm, and for 50 mm when the fixation screw was operational. The force was recorded as a function of time and hence distance into the sand and the geometrical parameters of the ensuing depression were measured as well (b) and (c).

Fig. 8. Calibration curves; (a) Force versus distance obtained for MSS-D. The slope S of the force versus distance is for small forces (< 50 N) Si ≈ 12.5 N/mm. For large forces (> 400 N) Se ≈ 60 N/mm; (b) For WF 34: Se ≈ 52 N/mm (large forces; > 400 N); (c) For small forces (< 10 N) the slope for WF 34 is Si ≈ 10 N/mm. In all 12 calibration tests carried out in MSS-D and WF 34, similar values for the slopes were obtained. In the six calibration tests in WF 34, the sand yielded suddenly at a force of ≈ 300 N, i.e. at a pressure of ≈ 32 kPa. However, this behavior was not observed in MSS-D. The initial force versus distance varies as K ≈ K0(δx –δ0,x)n with n ≈ 1.91 for WF 34 and n = 1.61 for MSS-D, see also text.

From these calculated geometrical data, one can estimate the bedding modulus MB introduced above MB ≈ Ko/(a2δx) = 21 MN/m3 for MSS-D. Values for loose sand are typically 80 MN/m3 (Kolymbas, 2011; Lang et al., 2011). However, one has to keep in mind that the bedding modulus depends also on the geometry of the penetrator and according to the standard DIN 18134 flat plates with diameters of 300, 600, and 762 mm should be used to measure it. Therefore, the value of 21 MN/m3 is within the range expected. Likewise, one can estimate the deformation modulus MEp = (dσ/dh)×d = Ko×a/(a2δx) ≈ 0.5 MN/m2 = 500 kPa (a = 2.4 cm; δx = 1.5 mm). These values are in reasonable agreement with tabulated values (Kolymbas, 2011; Lang et al., 2011). Let us estimate the strain stored in the surface area up to a depth of δx = 1.5 mm. Here, ∆ε = ∆δx/δx ≈ 1×10-3 = ((maximal deformation length)/(height of sand pile))× ((force at δx = 1.5 mm)/(maximal force)) = 9

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

15×18.7/(450×550), appreciably larger than the value when measuring CASSE time-of-flight data in sandy material where ε is at most 10-6 (Truell et al., 1969). For WF 34 the estimated contact-radius taken from the pictures was a = 2.8 ± 0.5 cm. With R = 200 mm, one obtains δx ≈ 2.0 ± 0.8 mm. Using Si ≈ 10 N/mm from Fig. 8c, the average measured K0 = 8.3 N corresponds to a penetration of δx ≈ 0.83 mm and a calculated contact-radius of a = 1.8 cm, still in reasonable agreement with the value expected, given all the inaccuracies. Furthermore, one obtains for WF 34 (K0 = 8.3 N with a = 1.8 cm and δx = 0.83 mm), MB ≈ Ko/(a2δx) = 30 MN/m3 and MEp = Ko×a/(a2δx) = 0.56 MN/m2 = 560 kPa. Both the bearing capacity and the deformation modulus for WF 34 are comparable to the values for MSS-D. In order to complete the analysis of the elastic/plastic properties of WF 34 and MSS-D, ultrasonic measurements were carried out, see appendix. We obtained for WF 34 the Young’s modulus E = 241 MPa, the shear modulus G = 108 MPa and the Poisson ratio υ = 0.12. For MSS-D the values are E = 469 MPa, G = 175 MPa, and υ = 0.34. The elastic and plastic data for MSS-D and WF 34 correspond to the results of previous measurements (Schröder, 2011) Furthermore, calibration curves were taken with the fixation screw operational. Here, vx = 1 mm/s. First, only the soles touched the surface. The slopes were in this part ≈ 11.2 N/mm (MSS-D), respectively ≈ 10.7 N/mm (WF 34). The screw as penetrator causes a smaller restoring force in sand, and this is indeed observed for the landing velocities vx = 1.1 m/s. Here K0,av = 8.3 N (MSS-D) and K0,av = 2.2 N (WF 34) instead of K0,av = 18.6 N and K0,av = 8.3 N at small vx, see above. The force versus distance curve can be fitted by a function of the form K ≈ K0,n(δx –δ0,x)n

(12)

Here δx is the depression and δ0,x is the coordinate where the sole touches the surface, and K0,n is the pre-factor in [N/mmn] for each n. The average values of the power dependence for the force versus depression δx are n ≈ 1.9 ± 0.3 for WF 34 and n = 1.6 ± 0.2 for MSS-D. With the screws in operation, the average values of the power dependence for the force versus depression δx are n ≈ 1.1 ± 0.3 for WF 34 and n = 1.62 ± 0.5 for MSS-D. Inserting numbers for δx into the fit equations reproduce the experimental findings. However, not much more can be derived from these expressions because the restoring force in sand are also strain-rate-dependent (Omidvar et al., 2012), calling for additional calibration experiments. They may be carried out when deceleration data from the landing are available. The fit curves, however, corroborate our analysis because the penetration dependence is sufficiently close to the δx3/2 dependence for elastic restoring force in Hertzian contact mechanics, see Eq. (10).

Fig. 9. Calibration curves, i.e. force versus distance obtained for MSS-D and WF 34 with the fixation screw operational. At first, when only the sole touched the sand’s surface, the initial slopes were Si ≈ 11.2 N/mm, respectively 10.7 N/mm, about the same values as obtained with the data shown in Figs. 8a and 8c. When the screw starts to penetrate, the mean slope of the force versus distance reduces to ≈ 1.8 N/mm, respectively 1.7 N/mm showing variations in the force as the screw penetrates deeper.

Summarizing this part, it is interesting to note that the effective forces are about the same for the landing tests on concrete or the two sands because of the action of the landing gear structure. 10

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

Restoring Forces Determined by the Inertial Mass of the Philae Foot When the fixation screws start to operate in the landing test for velocities larger than vx = 0.8 m/s, each of the three feet have a free distance of about 80 mm to travel in their guidance (Fig. A1b) before they are stopped, leading to the deflection of the landing gear. This corresponds to a time delay of ∆t = 80 ms for a landing velocity of 1 m/s. In this case the force acting on the sole is determined by the inertia of the sole’s mass and in addition by the Coulomb friction in the guidance system (Fig. 10). The total mass of a foot with the two soles is about mfoot ≈ 820 g. An average force of 20 N is necessary to accelerate this mass to a velocity of 1 m/s within 40 ms of time. In the LAMA tests, the fixation screw mechanism showed a high friction force (larger than 50 N) and therefore the screw only started to operate at landing velocities larger than 0.8 m/s. However, there is still an FS foot model available which was not used in any LAMA test so far. The frictional force in this unit was 2.4 N. Thus the total force would be ≈ 12 N. This value serves a clue for the forces in the FM model because this can no longer be measured.

Fig. 10. Mechanical equivalent model for the Coulomb friction force in the fixation screw mechanism in addition to the inertia force by the mass of a foot. In the LAMA tests, this force did not operate for landing velocities up to vx = 0.8 m/s, see text.

VII.

Sole Contact-Resonances

During development and qualification of CASSE, eigenmodes and eigenfrequencies of the soles were characterized. A detailed discussion of the results obtained can be found elsewhere (Arnold et al., 2004; Schieke, 2004). Furthermore, the free resonance-frequency of a flight spare sole with the accelerometer mounted showed two resonance frequencies, namely 600 and ca. 650 Hz (Krause, 2007). The double resonance is most likely caused by the mounting of the accelerometer lifting the degeneracy of the first eigenmode of the sole. This is typical for geometrical oscillators that are not completely symmetric and similar phenomena were observed for the sole’s higher eigenmodes (Arnold et al., 2004; Schieke, 2004). If one assumes that the soles were circular flat plates, clamped at its edges, and having a diameter of 2R and with a thickness t, the frequency f01 of the lowest eigenmode is given by (Graff, 1975; Morse and Ingard, 1968): f 01 = 0.9342

t R

'2

E 3ρ(1- υ 2 )

(13)

The diameter of the flat plate is equal to the diameter of the sole, hence 2R’ = 100 mm, t = 1 mm is the thickness of the sole, E = 20 GPa is the Young modulus of the glass-fiber epoxy material with a Poisson ratio υ = 0.3, and ρ = 2.3×103 kg/m3 is its density (Schwartz, 1992). Therefore, the lowest eigenfrequency should be f01 = 670 Hz, close to the observation given that the sole’s geometry is a clamped half-dome. It is also noteworthy that in the actual flight model a resonance frequency of 641 Hz was measured during the cruise phase (Seidensticker et al., 2012). From Fig. 2b, Figs. 6a and 6b one can see that the contact-resonances of the soles are excited by the landing shock both on the sands and when hitting a stiff material, here concrete. Apparently, the 11

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

landing shock acts like a δ-impulse having sufficient frequency bandwidth in order to excite eigenfrequencies of the soles. In four test landings on concrete with vx = 0.1 m/s comprising 12 data sets (one for each foot touching the surface), 11 could be analyzed and the sole contact-resonances were determined. In three cases foot # 2 touched the surface first. Three signals of two feet overlapped, however, always with a time delay larger than 50 ms relative to their onsets. Only the first 20 ms of the signals were analyzed for the data presented here. The contact-resonance frequencies varied between 1.2 kHz and 2.5 kHz. The highest values were measured at the largest peak deceleration values of bmax ≈ -1000 m/s2, and the lowest at the smallest bmax ≈ - 200 m/s2. The leading and trailing edges of the impulses showed lower frequencies, see Figs. 6. This means that the impulse is dispersive as a function of force acting on a foot. The data set from one test on concrete with vx = 0.2 m/s yielded still higher frequencies in comparison to vx = 0.1 m/s: fcr = 2.9 kHz (foot 1), fcr = 2.5 kHz (foot 2, see Fig. 6a)), and fcr = 2.7 kHz (foot 3). Also in this case, the impulses were dispersive with increasing force, see Fig. 6a. The data for the sole’s contact-resonances for the landing tests on MSS-D and WF 34 are summarized in Table II together with the data for concrete. No dispersion within a single pulse on either sand could be discerned due to the scattering of the data (Fig. 2b). However, as can be seen from the data, there is a clear tendency that the averaged contact-resonance frequencies for both sands increase with impact speed and hence static load like for concrete. Table II: Contact resonances of the soles

Material

Velocity [m/s]

Number of measurement

MSS-D MSS-D MSS-D MSS-D WF 34 WF 34 Concrete Concrete

= 0.1 Vx = 0.2 Vx = 0.5 Vx = 1.1 Vx = 0.1 Vx = 0.5 vx = 0.1 vx = 0.2

12 8 5 1 9 5 11 3

Vx

Average contact resonance frequency [Hz] 454 548 617 771 397 581 2003 2500

Minimal/maximal contact-resonance frequency [Hz] 334/525 461/683 414/744 240/500 435/650 1200/2500 2500/2900

In order to understand the behavior of the contact resonances, let us have a look at the basic behavior of a contact-oscillator. The angular contact-resonance frequency ωct of two freely supported bodies of mass m1 and m2 can be written as (Johnson, 1985): 2 ωct = k* ×

m1 + m 2

(14) m1m 2 where k* is the contact stiffness. The masses m1,2 are effective masses and depend on the geometry of the oscillator. If there is friction in the contact volume or in any other part of the oscillator, k* = kr + iki becomes complex. For viscous damping, the imaginary part of the contact stiffness is ki = ω×γ, where γ is the viscous damping (Ns/m) in the contact zone. This entails also a complex reduced modulus. There are a number of contact oscillators described in the mechanical engineering literature some time ago, where most of the aspects important for our case are discussed (Babitsky, 1998; Hess and Soom, 1991; Nayak, 1972; Sabot et al., 1998). In non-destructive testing the local mechanical impedance is measured using contact-resonances in the so-called Fokker bond tester (Guyott et al., 1986). When contacting a component, one measures with a Fokker bond-tester the shift in resonance frequencies of the oscillator, which depend among other parameters on changes of the adhesive properties of layers within the stress-field of the contactor, usually a sphere attached to the exciter. Typically, the exciter oscillates in the kHz range, and the corresponding wavelengths are in the decimeter range. Contact radii are in the millimeter range. More recently, contact-resonances techniques have been implemented in atomic force microcopies (Marinello et al., 2013; Rabe et al., 1996; Tian et al., 2004). Applying the content of these papers to our case here, the behavior of the contact-resonance frequen12

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

cies can be explained easily. During the first contact of the landing sole with either concrete or sand, the contact area increases with increasing time until the maximal deceleration is reached. In case of the elastic contact sole-concrete, this leads to a larger contact area and hence to an increase of the contact stiffness k* = 2aE*, which in turn causes a higher contact-resonance frequency. In the case of the test landings on sand, there are two effects: the mass loading increases with increasing time and with higher landing velocities, because a larger volume of sand is displaced. At the same time the sand consolidates more, and hence the deformation modulus and therefore the contact stiffness increases, which leads to an increase of the contact-resonance frequency like in the case of concrete. Attachment of masses to some of the experimental soles in the developing stages of CASSE indeed led to a decrease of their eigenfrequencies (Arnold et al., 2004). After the landing of Philae on 67P/Churyumov-Gerasimenko and provided that data of the contact-resonances of the soles are obtained, we will undertake finite element modeling of the sole oscillations in order to obtain quantitative values for the elastic/plastic contact stiffness parameters. VIII.

Summary and Future Work

Upon landing of Rosetta on 67P/Churyumov-Gerasimenko, it should be possible to determine from the accelerometer signals whether the comet soil is stiff, i.e. icy or softer, i.e. sandy. The measurement of the length Tc of the first landing deceleration signal, its amplitude b(ω) as well as the strength of the sole’s contact resonances and their frequency, should render possible that differentiation. In case the landing place consists of a material with a high elastic modulus in the GPa range, one cannot calculate its elastic modulus from the envelope of the landing shock because almost all deformation occurs in the landing gear. For softer materials, i.e. with a reduced modulus E* ≈ 1 MPa and less, at least an estimate of the elastic modulus should be possible from the measured b(ω) . If the sink-in of the lander craft could be measured in a sandy-like landing site, quantitative data for the bedding modulus, a plastic parameter, can be obtained. The detailed analysis of the contact resonances of the soles by finite-element techniques should allow to obtain more quantitative data on the elastic and plastic properties of the landing site in conjunction with the time-of-flight data of the sounding signals on the comet surface generated by the CASSE instrument (Kochan et al., 2000; Seidensticker et al., 2007). We should like to point out that the present analytical model is only an approximation of the situation we encountered in the landing tests or which we will encounter upon landing on the comet 67P/Churyumov-Gerasimenko. Neither are the restoring forces of the contact sole-surface material linear, nor is the time variation of the landing shock purely sinusoidal. Finally, the procedure described here can be adapted to other spacecraft landings on objects of the Solar system. Acknowledgement

One of us (W. A.) thanks K. Samwer, 1. Phys. Institute, Göttingen University and J. Turner, Department of Mechanical Engineering, University of Nebraska, Lincoln, NE, for helpful discussions. He also thanks D. Bruche, Fraunhofer IZFP, Saarbrücken, Germany, for help in the measurement of the elastic moduli of WF 34 und MSS-D by ultrasonic techniques. Last but not least W. A. thanks H. Bentaher from the Higher Institute of Industrial Systems, Gabes, Tunesia, for helpful discussions on soil mechanics.

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Appendix 1. Estimate of Stiffness of a Rosetta Lander Foot for Vertical Loading

(a)

(b)

Fig. A1. (a) Accelerometer sole mounted in a Philae foot with its lid opened. The sole contains an electrode of the permittivity probe (PP) with a preamplifier mounted into the inside of the lid. The accelerometer is fixed between glass-fiber clamps in the center of the sole (from (Seidensticker et al., 2007); (b) Assembled feet attached to the landing gear. The left sole houses the transmitter piezoelectric element, and the right foot the accelerometer shown in (a);

Fig. A2. Design of the foot sole. It consists of three layers of a glass-fiber epoxy composite. The sole is bonded to an aluminum ring having a diameter of 100 mm. The lid of the sole is attached by an adhesive to the Al-ring as well. The central part of the lid is stiffly attached to a rectangular tube which is guided by a fixture holding the suspension and mechanical control of the fixation screws in the landing gear; see Fig. A1(b).

(i) Estimate of the sole’s stiffness according to Roark’s Formulas for Stress and Strain (Young and Budynas, 2002): Treatment of the sole as a thin-walled pressure vessel, i.e. as a partially spherical shell of opening angle φ = 90°, case 2 on page 610 in (Young and Budynas, 2002). The partial shell is symmetrically loaded by the force P over a central part, given by the contact radii discussed above. 14

Submitted to Planetary and Space Science, September 5th, 2014, revised November 5th, 2014

The stiffness is then given by ksole = Et2/(R(1-υ2)0.5A1). Because the shell is vertically supported by the aluminum ring and hence guided, the geometrical parameter is A1 = 0.36 (Young and Budynas, 2002). The other required parameters are E = 20 GPa (glass-fiber epoxy material), with an assumed υ = 0.3, t = 1 mm (thickness of the shell), and R = 250 mm. One obtains ksole ≈ 233 kN/m. (ii) Estimate of the lid stiffness: Assuming that the shape of the lid can be approximated as a spherical segment of radius R ≈ 340 mm which follows from its diameter Ø = 100 mm and its height ∆h = 3mm, one obtains according to the same reference, page 610, again case 2, klid = Et2/R(1-υ2)0.5A1). In this case A1 ≈ 0.31 (Young and Budynas, 2002). For aluminum E = 69 GPa, υ = 0.34, and t = 1 mm. This yields klid ≈ 696 kN/m. This is a lower limit because of the fixture device in the center of the lid making it stiffer. (iii) Stiffness of the glass-fiber ring connecting the sole to the aluminum ring and also holding the lid: kr = Et (page 592 of ref. (Young and Budynas, 2002)). With E = 20 GPa and t = 1 mm, this yields kr = 20 MN/m. (iv) Stiffness of the aluminum ring: This is a more complicated case to calculate. The ring serves as radial stiffener for loads acting on the foot and the lid. Let us assume that the force acting on the sole is transferred radially to the ring. Then, k = 2πEty/R, where R is the ring radius, t is its thickness, and y is its height, here y ≈ t = 8 mm (read from the above figure). For aluminum EAl = 70 GPa, kAlr ≈ 563 MN/m. This large value justifies the assumption of a clamped rim for the sole ring and for the sole lid, in order to calculate their stiffnesses according to (i) and (ii). The total stiffness is then given by: 1/ksole = ∑1/ki which leads to ksole ≈ 173 kN/m, which is much larger than the stiffness of the landing gear. The total stiffness is dominated by the lid as well as by the sole. Here, it should mentioned that a measurement of the stiffness of the landing gear during its development stage, with and without the feet attached, yielded the same value within measurement accuracy, corroborating the above estimates (R. Roll, 2014, private communication). 2. Measurement of the Elastic Moduli of the Sands WF 34 and MSS-D

To measure the elastic moduli of the sands, ultrasonic time-of-flight measurements were carried out, either through transmission or in back reflection. Path lengths in the sands varied from 5 mm to 16 mm. The time-of-flight data yielded a sound velocity for longitudinal waves of vl = 390 m/s and vs = 257 m/s for shear waves for WF 34, and respectively vl = 462 m/s and vs = 321 m/s for MSS-D. We measured the sand aggregates densities and obtained ρ = 1.64×103 kg/m3 for WF 34 and ρ = 1.70×103 kg/m3 for MSS-D. Both the density and sound velocities were pressure-dependent. A weight of 10 kg was used to maintain a firm contact of the transducer (Panametrics Videoscan V1012, nominal frequency 250 kHz for longitudinal waves and Panametrics V 153, nominal frequency 1 MHz for shear waves) with the sands’ surface. The diameters of the transducers were 40 mm and hence the weight amounted to a static pressure of ≈ 80 kPa with which the sands were loaded. From the sound velocities and the densities, it is straightforward to calculate Young’s modulus E, the shear modulus G, and the Poisson ratio for both aggregates, using well-known equations (Truell et al., 1969) and which are also the basis for the CASSE experiment (Kochan et al., 2000; Seidensticker et al., 2007). We obtain for WF 34 E = 241 MPa, G = 108 MPa and υ = 0.12 and for MSS-D E = 469 MPa, G = 175 MPa, and υ = 0.34. Note that the sand elastic modulus depends on the consistency and packing (density) of the soil. Without static pressure the densities for WF 34 and MSS-D were 1.5×103 kg/m3 and the longitudinal velocities were 326 m/s, respectively 365 m/s for WF 34 and MSS-D. However, shear waves could not be generated if the transducers were not pressed into the sand. From the variability of the time-flight measurement, it is estimated that numbers for E and G are accurate within 30%. The data we obtained agree with values in tables for dense sands. The corresponding strains ε in ultrasonic measurements are very small, less than 10-6, so that one indeed measures elastic quantities, here E and G (Truell et al., 1969) and not a plastic quantity, see also the discussions above. Finally, note that the velocities in the sands are higher than in comet analog material (Kochan et al., 2000).

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References

Arnold, W., Gebhardt, W., Licht, R., Kochan, H., Schieke, S., 2004. Contributions of the Fraunhofer Institute for Non-Destructive Testing to the Rosetta Lander Instrument CASSE: Design, Calibration, Analysis. Fraunhofer IZFP report . No. 040211-TW, Saarbrücken, pp. 61. Ashby, M., et al., 2009. Engineering Materials and Processes Desk Reference. Elsevier, Amsterdam. Babitsky, V. I., 1998. Theory of Vibro-Impact Systems and Applications. Springer, Berlin. Biele, J., Ulamec, S., 2008. Capabilities of Philae, the Rosetta Lander. Space Science Reviews. 138, 275-289. Biele, J., Ulamec, S., Richter, L., Knollenberg, J., Kuehrt, E., Moehlmann, D., 2009. The putative mechanical strength of comet surface material applied to landing on a comet. Acta Astronautica. 65, 1168-1178. Graff, K., 1975. Wave Motion in Elastic Solids. Oxford University Press, London. Guyott, C. C. H., Cawley, P., Adams, R. D., 1986. The Non-Destructive Testing of Adhesively Bonded Structure-A Review. Journal of Adhesion. 20, 129-159. Hashin, Z., Monteiro, P., 2002. An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste. Cement and Concrete Research. 32, 1291-1300. Hess, D. P., Soom, A., 1991. Normal Vibrations and Friction under Harmonic Loads. 1. Hertzian Contacts. Journal of Tribology-Transactions of the Asme. 113, 80-86. Hobbs, P. V., 1974. Ice Physics. Clarendon Press, Oxford, UK. Johnson, K. L., 1985. Contact Mechanics. Cambridge University Press, Cambridge UK. Kochan, H., et al., 2000. CASSE - The ROSETTA Lander Comet Acoustic Surface Sounding Experiment - status of some aspects, the technical realisation and laboratory simulations. Planetary and Space Science. 48, 385-399. Kolymbas, D., 2011. Geotechnik. Springer, Heidelberg. Kömle, N. I., et al., 1997. Using the anchoring device of a comet lander to determine surface mechanical properties. Planetary and Space Science. 45, 1515-1538.. Krause, M., 2007. Untersuchung von Vibrationen an Bord der ESA-Mission mit dem Instrument CASSE. Fachhochschule, Aachen, Germany. Lang, H.-J., Huder, J., Amann, P., Purin, A. M., 2011. Bodenmechanik und Grundbau. Springer, Heidelberg. Marinello, F., Passseri, D., Savio, E., 2013. Acoustic Scanning Probe Microscopy. Springer, Heidelberg. Meyer, E., Guicking, D., 1974. Schwingungslehre. Vieweg Verlag, Braunschweig. Morse, P. M., Ingard, K. U., 1968. Theoretical Acoustics. Princeton University Press, Princeton. Nayak, P. R., 1972. Contact Vibrations. Journal of Sound and Vibration. 22, 297-322. Omidvar, M., Iskander, M., Bless, S., 2012. Stress-strain behavior of sand at high strain rates. International Journal of Impact Engineering. 49, 192-213. Rabe, U., Janser, K., Arnold, W., 1996. Vibrations of free and surface-coupled atomic force microscope cantilevers: Theory and experiment. Review of Scientific Instruments. 67, 32813293. Sabot, J., Krempf, P., Janolin, C., 1998. Non-linear vibrations of a sphere-plane contact excited by a normal load. Journal of Sound and Vibration. 214, 359-375. Schieke, S., 2004. Beiträge zur numerischen Simulation des Instruments CASSE der ESA Rosetta Mission. Naturwissenschaftlich-Technische Fakultät III, PhD thesis. Saarland University, Saarbrücken. Schröder, S., 2011. Soil Characterization Report. DLR Bremen. No. LSD-TN-04-DLR , Bremen, pp. 11. Schulz, R., Alexander, C., Boehnhardt, H., Glassmeier, K.-H., 2009. ROSETTA ESA's Mission to the Origin of the Solar System. Springer. Schwartz, M., 1992. Composite Materials Handbook. McGraw-Hill, Inc., New York. Seidensticker, K. J., Fischer, H.-H., Schmidt, W., Hirn, A., 2012. Philae Payload Checkout 13, Cruise Phase Report. No. RO-LSE-RP-3102. DLR, Cologne, Germany, pp. 27. Seidensticker, K. J., et al., 2007. SESAME - An experiment of the Rosetta Lander Philae: Objectives and general design. Space Science Reviews. 128, 301-337. 16

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Spohn, T., et al., 2009 MUPUS-The Philae Thermal and Mechanical Properties Probe. In: R. Schulz, C. Alexander, H. Boehnhardt, K.-H. Glassmeier, (Eds.), Rosetta ESA's Mission to the Origin of the Solar System. Springer, New York, pp. 651-667. Tian, J., Ogi, H., Hirao, M., 2004. Dynamic-contact stiffness at the interface between a vibrating rigid sphere and a semi-infinite viscoelastic solid. Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control. 51, 1557-1563. Truell, R., Elbaum, C., Chick, B., B., 1969. Ultrasonic Methods in Solid State Physics. Academic Press, New York. Witte, L., et al., 2014. Experimental Investigations of the Comet Lander Philae Touchdown Dynamics. Journal of Spacecraft and Rockets. 1-10. Witte, L., et al., 2013. Philae-Lander Touchdown Dynamics Revisited-Tests for the Upcoming Landing Preparations. In: P. Papadopoulos, J. Cutts, (Eds.), International Planetary Probe Workshop Archive (IPPW-10). NASA. Wong, J. Y., 1980. Data Processing Methodology in the Characterization of the Mechanical Properties of Terrain. Journal of Terramechanics. 17, 13-41. Young, W. C., Budynas, R. G., 2002. Roark’s Formulas for Stress and Strain. McGrawHill, New York.

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Parameter Stiffness of a single leg of the landing gear Mass of the lander Damping constant on the lander structure Stiffness of the suspension of the lander structure

Symbol

Value

kLG

13.3 kN/m

mL

98 kg

γD

601 Ns/m

kLT

4.1 kN/m

Table I: Measured parameters of the lander model (Witte et al., 2014)

Material

Velocity [m/s]

Number of measurement

MSS-D MSS-D MSS-D MSS-D WF 34 WF 34 Concrete Concrete

= 0.1 = 0.2 Vx = 0.5 Vx = 1.1 Vx = 0.1 Vx = 0.5 vx = 0.1 vx = 0.2

12 8 5 1 9 5 11 3

Vx Vx

Average contact resonance frequency [Hz] 454 548 617 771 397 581 2003 2500

Minimal/maximal contact-resonance frequency [Hz] 334/525 461/683 414/744 240/500 435/650 1200/2500 2500/2900

Table II: Contact resonances of the soles

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Highlights: •

Inverting Philae’s deceleration signals to measure surface elastic/plastic properties of 67P

•

Exploitation of the foot-sole’s contact-resonances to obtain elastic/plastic surface data of 67P

•

Calibration of the deceleration signals on stiff and soft materials

19