Nonlinear Decelerator for Payloads in Aerial Delivery Systems. I: Design and Testing
T. Lyons, M. Ginther, P. Mascarenas, E. Rickard, J. Robinson, J. Braeger, H. Liu, A. Ludu
aa r X i v : . [ c ond - m a t . s o f t ] A ug Nonlinear Decelerator for Payloads in Aerial DeliverySystems. I: Design and Testing ∗ T. Lyons, M. Ginther, P. Mascarenas, E. Rickard, J. Robinson, J. Braeger, H. Liu, and A. Ludu † Department of Mathematics, Embry-Riddle Aeronautical Univeristy,Daytona Beach, FL 32114, USA Adams State University, Alamosa, CO 81101, USA Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA
AbstractWe study the dynamics and the optimization of the shock deceler-ation supported by a payload when its airborne carrier impacts theground. We build a nonlinear elastic model for a container proto-type and an elastic suspension system for the payload. We model thedynamics of this system and extract information on maximum de-celeration, energy transfer between the container and payload, andenergy resonant damping. We designed the system and perform labexperiments for various terminal velocities and types of grounds (ce-ment, grass, sand water, etc.). The results are compared with thetheoretical model and results are commented, including predictionsfor deceleration at different types of ground impact. The resultscan be used for aerial delivery systems, splash-down of capsules,recoveries, weather balloons, coastal surveying systems, or the newintroduced goal-line technology in sport competitions. ∗ Preprint available at arxiv.org/submit/1045587/2014 , submitted to
Nonlin. Dyn. † Correspondence should be addressed to [email protected]. Introduction
Aerial delivery systems, [1], can be described as airborne compact containers, sometimesattached to a parachute, containing a payload consisting in analog and digital recordingequipment for collecting data at high altitude. Such systems can stream the recordeddata into a satellite network, or just store it and aim for further recovery on the ground.Given the fact that such system are in general expensive, and the data stream consists ina very large volume sent at very high frequency (like multi-spectral high definition videorecordings), recovering these systems is a desired procedure.In general, after accomplishing the data recordings, the air delivery systems are dis-connected from their aerial carrier, and released airborne with a parachute. Even in thissituation the ground impact can be seriously damaging for the sensitive payload, espe-cially since it is difficult to determine the landing on a specific predestinated ”soft” spot.The study of the maximal decelerations induced in the payload at their ground impactrepresents an interesting field of study. Moreover, a large spectrum of ingenious mechani-cal systems can be used to suppress these decelerations to an acceptable limit, no matterof the type of landing, be it on concrete, grass or water, etc.The system container plus payload can be design to minimize this ground impact de-celeration by using combinations of nonlinear normal modes from special elastic systems.One interesting solution is to use the non-linear energy sink process (also called targetedenergy transfer) in order to transfer quickly, by transient resonance capture, the shockenergy from the payload back to the container oscillations [2, 3].Such a design capable of quickly damping the payload deceleration can be further usedin a variety of different research projects including weather balloons, coastal surveyingsystems, goal-line technology in sport competitions, splash-down of capsules, etc.In this paper we present an air delivery system prototype capable of reduced payloaddeceleration touchdown for a variety of ground types and terminal velocities. Several suchsystems have been designed and tested thus far. The goal of this research is to study whatis the best protection system of the payload inside the container against the mechanicaldamaging effects of the high value of deceleration resulting from the ground impact.Section 2 begins by providing a brief description of the system and measurementmethods. In continues by describing the design of the container and the payload, and themathematical model. The container is analyzed in parallel by two methods: thin shellmodel modes of oscillations, and Hertz elastic deformation. The first model generatesthe normal frequencies of vibrations, and the second model generates a nonlinear forceof deformation, including a phase-transition. Exact solutions of the model equation foroscillations under this nonlinear deformation force are compared with the thin shell modesin order to obtain a reliable combined model for the elastic behavior of the container. Thepayload is also modeled by a system of springs whose force combines into a nonlinear in-teraction. The model equation for this force is also integrable, but the oscillation obtainedare too tedious to be studied exactly. We use a series expansion in order to calculate witha reasonable approximation the frequency of oscillation of this nonlinear force.In section 3 we present the theoretical model and its performances. We also comparethe theoretical results with the experiment and calculate rates of dissipation of energy.In section 4 we analyze the results of the experiments on drops performed at differentterminal velocities on different types of ground. The paper ends with conclusions andrecommendations for the further development.2
Container and payload
In order to understand the dynamics of the impact between the compound system con-tainer plus payload with the ground, and to predict maximal decelerations for differentterminal velocities and different types of ground we performed controlled experiments offree drops in the Wave Motion Lab at ERAU [4]. For the typical range of container sizesused in aerial data acquisition, and corresponding parachutes, the terminal velocities innormal conditions ranges in V ∼ − Hydroplus Engineering has an oblatespheroidal shape having its polar axis (along the vertical direction) 2 R k = 0 . m and itshorizontal diameter 2 R ⊥ = 0 . m . The container is made of two symmetric halves oftransparent Acrylite GP material of thickness h = 4 .
78 mm interconnected by a boltedFigure 1: View of the opened container during reading data after one drop. Four radialsprings connect the container’s wall to the cex ntral payload which contains the videocamera, accelerometers and the integrated circuit board. One accelerometer is placed onthe container’s wall, too. 3im through a water proof gasket of width 0 . m outstanding the spheroid surface. Ontop of its upper half the container has an drag-producing empennage made of four verticalfins connected by a horizontal circular stabilizer, see Figs. 1, 2. In order to model the container impact we need to study first the elastic properties ofthe container. One way to perform this analysis is to use the modal response of a thinspherical shell [5, 6, 7]. The axisymmetric vibration modes spectrum is basically composedof three countable sets of frequencies: one for the membrane, one for the flexural, andone higher mode. The non-axisymmetric modes have the same frequencies except in adegenerate way. The modal frequencies for the thin spherical shell are given by [5] f i = λ i πR s Eρ (1 − ν ) , i = 1 , , . . . , (1)where R is the sphere radius, E is the modulus of elasticity, ν is the Poisson ration and ρ is the density. The parameters λ i are given by the roots of the shell model equation [8]: α i λ − β i λ + δ i λ − γ i = 0 . (2)The coefficients for this equation depend on a mode number i , on the size h, R throughthe coefficient h / R , and on ν . In the case of our material we have ρ = 1 . · Kg/m , E = 4 . · N/m , ν = 0 . R = R k and h as above. The frequencies ofthe first modes are shown in Table 1, where the three rows represent the three doublereal roots of Eq. (2) The main modes of vibrations of the container have periods in therange 35 −
45 ms for the flexural modes (the first row in the table), and 5 . −
15 msfor the membrane modes (the second row), while the highest modes (third row) cannotbe excited by the impact initial conditions and at energies in the range of the impactwith ground. Studies of correlations of theoretical natural frequencies of spherical shellswith finite element simulations and with time-averaged dynamic holography experimentswith real spheres with imperfections show, [8], that the non-axisymmetric modes splittingtowards higher values of frequency, and the most likely modes of excitation are the shortperiod membrane modes.Another way of studying the elastic deformation of the bottom of the container is touse the Hertz elastic hypothesis, [9], where energy of deformation given by U = E Z (cid:18) ∂u∂z (cid:19) dV, (3)where u is the strain field and dV is the volume element. This approach seems to beappropriate for deformations induced only by bottom impacts where the local strains arerather one-dimensional induced by the vertical motion of the spheroidal container. Thishypothesis is also supported by a multitude of rapid photography data and measurementsof rotation and lateral acceleration. Indeed, since the bottom of the container, and thepayload suspension are built symmetrically off-axis lateral motion and rotations duringthe free fall and impact phases with a plane surface phases are very unlikely.4igure 2: Impact after H = 2 . h of the shell in average. In any of the phases shown inFig. 3 the elastic deformation energy has two terms. One term represents the compressionof the bottom spherical cap into a flat area (type I), or into a concave trough shape (typeII). The other term describes the bending, i.e. the occurrence of a circular fold aroundthe compressed domain. If we denote the axial deformation along the symmetry axis by ǫ << R , the compression part of the elastic deformation energy Eq. (3) can be evaluateby simple trigonometric calculations [10] U comp = πEhǫ R . (4)For the bending part of the elastic energy we use the theory of equilibrium of thin plates,[9], and following the Willmore energy formula, [11], we obtain U bend = Eh Z (cid:18) h H − ν ) + ¯ κK (cid:19) dA, (5)where H, K are the mean and Gaussian curvatures, dA is the area element, and ¯ κ is theGaussian curvature modulus [12]. In the limit of thin shell model, from Eqs. (4,5) weobtain the elastic energy U = U comp + U bend in the form [10] U ( ǫ ) = ( U I = Eπh / ǫ / − ν ) ] / R + πEhǫ R < ǫ ≤ ǫ c ,U II = Eπh / ǫ / − ν ) ] / R + πEh ǫ R ǫ c < ǫ, (6)where ǫ c is the smallest positive solution of the equation U I ( ǫ ) = U II ( ǫ ), and it representsthe point of phase-transition from the flat deformation to the trough deformation. Thesecond row in Eq. (6) differs from the first by a factor of 4 in the first term, and the lineardependence on ǫ in the last term.For small deformations the expression I is energetically favorable. However, as thedeformation increases U II < U I and a first order phase-transition arises at this criticaldeformation roughly proportional to h . The hysteresis associated to this transition (theforce has a jump) is strongly dissipative because of the friction work generated by thesliding of the contact point between the circular fold and the ground, when the radius ofthe fold changes. For a friction coefficient µ between the container material and ground,the dissipated energy can be approximated with W = Eµπh / ǫ R (1 − ν ) ] / , (7)showing is a linear dependence with the deformation.6nother interesting feature of the Hertz model is that we can estimate the impactFigure 3: Configurations of small axial deformation of a thin spherical shell of radius R at impact on a rigid horizontal plane. Left: (I) Flat deformation of height ǫ . Right: (II)Fold and trough deformation occurring usually at double amounts of deformation thanthe flat one. At higher impact force the bottom of the sphere buckles upwards and buildsan inversion of curvature. 7ime from the impact velocity with pretty good accuracy τ = (cid:18) | V | c (cid:19) / R hc , (8)where c is the speed of sound in the wall material and V is the initial impact velocity. Inour case we measured c = 2 ,
745 m/s, and for example for an impact at V = − τ ∼
62 ms, which is in good agreement with our measurements ofthe impact, and with the dynamical estimation, see the following sections.The elastic impact forces obtained from Eqs. (6) have the form F ( ǫ ) = (cid:26) − a √ ǫ − bǫ , ≤ ǫ ≤ ǫ c , − a √ ǫ − c, ǫ c < ǫ, (9)with the positive constants a, b, c obtained from Eqs. (6).In the absence of damping the dynamics induced by each of these two types of forces isexact integrable through F hypergeometric functions. For the type I force, the resultingnonlinear oscillator model mZ ′′ = − a √ Z − bZ , with initial conditions Z (0) = 0 , Z ′ (0) = − V has an implicit exact solution in the form t = 3 mZ (1 − Q Z / )(1 − Q Z / ) F (cid:18) ; , ; ; Q Z / , − Q Z / (cid:19) V m − aZ / − bZ , where F ( a ; b , b ; c ; x, y ) is the Appel hypergeometric function of two variables, and Q , = b − a ± p a + 3 bmV / . For the type II force, the corresponding nonlinear oscillator model mZ ′′ = − a √ Z − c ,with initial conditions Z ( t c ) = Z c , Z ′ ( t c ) = − V c has an implicit exact solution in the formΞ Z Z Z (cid:20) Ξ E (cid:18) sin − q Z Ξ (cid:12)(cid:12)(cid:12)(cid:12) Ξ Ξ (cid:19) − Ξ F (cid:18) sin − q Z Ξ (cid:12)(cid:12)(cid:12)(cid:12) Ξ Ξ (cid:19)(cid:21) Ξ Ξ (cid:18) C − cZ − Z / (cid:19) = ± t + C √ m , where F ( ·|· ) , E ( ·|· ) are the complete elliptic integrals of the first and second kind, re-spectively, and C , are constants of integrations. We define the symbols Ξ ij = Ξ i − Ξ j , Z i = p Z ( t ) − Ξ i , where Ξ i , i = 1 , , − C + 3 c Ξ + 8 a Ξ = 0.In Fig. 4 we present a numeric example. The elastic potential energy U from Eq.(6), the dissipated energy W from Eq. (7), and the resulting force F from Eq. (9)are plotted together with some exact oscillation solutions for various initial positions Z , Z ′ = 0, for a m = 4 Kg container made from the specified material. The jump in the8orce, and non-differentiability of energy are visible at the phase transition point whichoccurs at ǫ c = 0 . Z < ǫ c , the period of oscillations does not change too much with initialconditions. However, for Z > ǫ c the signature of the nonlinearity is noted through thestrong dependence of the oscillation period with initial conditions.From our fast photography experiments of dropping on concrete, and for heights H =1 ÷ Z or ǫ does not exceed 20 mm which proves that the ǫ c phasetransition limit is not reached for this type of situations. Comparing the elastic energyand friction dissipation from Fig. 4 at this maximum values of deformation with the initialmechanical energy in the drop 39 .
24 J at H = 1m, and 58 .
86 J at H = 1 .
5m reveals thefact that only half or less of mechanical energy is lost friction with the ground, i.e. 35%energy loss for H = 1 m drop, and 42% energy loss for drop at H = 1 . W H Ε L U II H Ε L U I H Ε L F I H Ε L F II H Ε L Z H t; Z L - - - t @ s D Ε @ m D Figure 4: The elastic energy U (solid curve, Eq. (6)), the friction energy W (dashedline, Eq. (7)), and the force F scaled 1 : 100 (dotted, Eq. (9)). In the inset we presentexact oscillation solutions for this force, for V = 0 and various initial positions Z : solidcurves for Z < ǫ c , and dashed curves for Z > ǫ c . The two horizontal solid lines in theinset represent Z = ± ǫ c . In spite of the discontinuity in force, the solution is smooth.Nonlinearity is noted in the dependence of the oscillation period with initial conditions.The dashed horizontal line is the maximum impact force measured by the acceleration ofthe container. 9nergy is lost in visco-elastic deformation of the container material which will be considerin the following. The value of the impact force measured by the accelerometer attachedto the container shows for H = 1 . ǫ = 0 . ǫ < ǫ c ∼ . . th − th mode predicted by the thin shell model, see the table above. Smalldifferences from the thin shell model, and the tendency to excite higher shell mode maybe related to the internal viscous and frictional forces in the container material whichwere not taken into account in the thin shell model. Also, for deformations larger than ǫ ∼ .
005 m the quadratic term in the Hertz force of type I in Eq. (9) ( F I ) is dominantover the square root. In this case, the solution for the oscillations mZ ′′ ∼ − bZ is givenby the Weierstrass elliptic function ℘ (( bZ/ / ; 0 , T ) which provides smaller values forthe period of oscillations (stronger interaction). The payload consists in a square shaped solid box containing electronics, sensors, batteriesand a video camera pointing downwards, see Fig. 1. It is suspended at the center of thecontainer by four linear springs of equilibrium length l and elastic constant k which allowit to oscillate up and down inside the container, see Fig. 2. If we denote by z this verticaldisplacement relative to the container, the total vertical force has the expression f ( z ) = − kz p z + l − l p z + l , (10)where l would be the extension of each spring when a massless payload stays in equi-librium in the equatorial plane. This nonlinear force has a linear term if the springs arepre-tension, that is if l > l . The series expansion of the nonlinear force is f ( z ) = − k (cid:18) − l l (cid:19) z − k l l z + 3 k l l z + O (cid:18) zl (cid:19) . (11)and f ( z ) → − kz when z → ∞ . The oscillations generated by this nonlinear force on amass m with initial conditions z (0) = 0 , z ′ (0) = − V , K = mV / t = C ± r m Z z ( t )0 ds q − kl − ks + 4 kl p s + l . (12)The RHS term in Eq. (12) is reducible to a sum of two elliptic integrals (on of the firstkind and one incomplete) and the nonlinear oscillation can be expressed analytic in termsof special functions. In order to estimate in a first order of approximation the frequencyof these oscillations we approximate the force with its cubic Taylor polynomial and weobtain a good approximation for the cubic nonlinear oscillations frequency of the payload10nside the container ν [ Hz ] ∼ p kl (2 l − l ) / z (cid:20) l − l ) + ( l − l ) z l (cid:21) . (13) In this section we write the dynamical equations for the system container plus payloadunder the action of the gravity, impact with ground modeled by the elastic force of thecontainer Eq. (9), interaction between payload and container modeled by Eq. (10), airresistance forces, and internal friction forces. We consider the motion 1-dimensional with2 degrees of freedom. For a container of mass M and a payload of mass m we have thedynamical system M Z ′′ = − M g − F ∗ ( L − Z ) − f ( z ) − AZ ′ ∗ − △ ∗ sign( Z ′ ) − BZ ′ ∗ sign( Z ′ ) − B a Z ′ sign( Z ′ ) , (14)( M + m ) z ′′ = − mg + f ( z ) − αz ′ − δ sign ( z ′ ) , (15)where Z ( t ) is the height of the center of mass of the container with the origin taken onground. By L we denoted the distance between the center of mass of the container and thelowest contact point of the container, so this is the height at which the impact force startsto act upon the system. The parameters A, B, △ are positive constants describing thedrag coefficients for Stokes linear viscous force, Rayleigh quadratic viscous, and constantfriction forces responsible of the container deformation. The star superscript shows thatthe quantity has that value only during the mechanic contact with the ground, and it iszero otherwise, if the container is airborne (that is ( x ) ∗ = x if 0 ≤ Z ≤ L and is zeroelsewhere). The parameter B a = ρ air C D πR ⊥ / − m/s with parachute, and 24 − m/s without. Inboth situations however, the Reynolds number ranges between Re = 2 , − , t ≥
0, where 0 here denotes the impact moment, under initial conditions Z (0) = L , z (0) = z eq , Z ′ (0) = − V < , z ′ (0) = 0. Here z eq is the equilibrium position of thepayload inside the container at rest, that is the solution to the equation f ( z eq ) = − mg .This equation can be reduce to a quadric equation z + mg k z + (cid:18) m g k + l − l (cid:19) z + mgl k z + m g l k = 0 . z eq for this equation, and it canbe approximated with z eq ∼ − mg k (cid:18) s − k ( l − l ) m g (cid:19) + O ( mg/k ) . The linearization of the differential system around the equilibrium values Z ′′ = z ′′ , Z ′ = z ′ = 0 , F ( Z eq − L ) + f ( z eq ) = − M g, f ( z eq = − mg. conducts to an eigenvalue algebraic equation of the form − kǫM + (cid:20) λ ( α + λ ) + mM (cid:18) mM (cid:19) km (cid:21)(cid:18) ǫM + Aλ + λ (cid:19) = 0 . (16) Drop heighta = =- g Equilibrium positionZero force levelContainer motionAcceleration x - Impact
Total energy
Gravitational energyElastic energyKinetic energy
Energy balance @ s D @ s D Figure 5: Model calculations for impact of an empty container on cement. Lower curveshows motion of the sphere few milliseconds before the impact, and upper curve shows itsacceleration (shifted upwards and re-scaled in this figure). In the inset we present energybalance between potential energy stored in the elastic force and gravitational and kineticenergy. The sum of all these energies is drawn with a thicker curve and shows two dropsin energy: first steeper one by the visco-plastic impact deformation, and the second dropby visco-elastic deformation of the container’s wall.12he analysis of the eigenvalues is performed in Figs. 6, 7, and 8.In order to test the model we integrated numerically Eq. (14) for m = 0. The resultis presented in Fig. 9.The motion of the container (sphere) Z ( t ) a little before the impact, and after theimpact and its acceleration are calculated. The energy balance shows a two-phase processof impact: in the first 15 ms after impact the damping is dominated by the visco-plasticimpact deformation, and the acceleration picks to its maximum value in the whole process.In the second stage, the process is longer (60 ms) and the damping is dominated by regularfriction forces which burn the energy already stored in elastic force.In Fig. 9 we present the theoretical calculations compared to the experimental results.In this case the container has M = 2 . Kg out of which the payload was m = 0 . Kg ,and it was dropped from H = 2 . m on concrete. The best model parameter fit wereobtained for k = 740 N/m, K = 61 , N/m , α = 5 . Kg/s, A = 4 . Kg/s, δ = 11 . N ,and ∆ = 51 . N .The relative errors for accelerations are smaller because of the precision of the elec-tronic sensors, while the errors in the visual measurement of the container position slightlyincrease towards higher times because of the parallax error.The experiment-theory match is good enough to induce confidence in the model, andan analysis of the stages of the impact process can be made. In the first stage (labeled I in Fig. 10) the container hits the ground, and its center of mass keeps descending a shortinterval of time because of the walls elasticity. In this stage the container experienceshigh acceleration with a maximum of about 600 g . At the same time, the payload begins - - - - - Λ real part Λ i m a g i n a r yp a r t - - - - - Λ real part Λ i m a g i n a r yp a r t Figure 6: Eigenvalues for the linearized system of differential equations for a set of pa-rameters ranging k = 10 N/m, M = 0 .
55 Kg, m = 0 . α = 0 . ÷ · − ÷ . ÷ . A = 0 ÷ △ = δ = 0 N. 13ts trip downwards, also with high downwards acceleration, phase labeled II , see also Fig.2 (a). In this second phase, even if the container tends to bounce off the ground, itsupward tendency of (see label II on top of the ”Container position” solid curve) motionis slowed down by the springs opposite reaction. The acceleration of the payload fullydevelops, and the container is maintained at ground for a while. After experiencing thislarge value, the payload acceleration changes the sign, and the payload moves upwards.The payload stops at a highest upper point, marked by III in Fig. 10, and also noticedin the frame (b) of Fig. 2. Simultaneously, the container starts to feel the spring effectand is pulled upwards, too, see label
III under sphere acceleration solid gray curve, Fig.2 (c). In phase IV the container is lifted from ground at its maximum bouncing height,while the payload is in opposition of phase: its acceleration is oriented downwards, seeframe (d) in Fig. 2. The container bounces maximum twice times, while the payloadperforms 2.5 full oscillations. We notice that for the chosen configuration of the elasticcoefficients and masses the damped container oscillations are somehow in opposition ofphase with the payload, as one can see from the upper part in Fig. 10 where the twocurves representing their motion have always maxima in opposition. In Fig. 11 we presentthe energy balance in one typical impact. An interesting observation must be mentionedbecause the phenomenon repeats almost in all situations. The first acceleration stageright after impact the container and payload accelerations are in opposition of phase, andthey attain their maximum accelerations at minimum total kinetic energy, and maximumtotal potential energy, see Fig. 12, too. The study in Fig. 12 describes a container ofradius R = 0 .
127 and mass M = 1 Kg, with a payload m = 0 . l = l = 0 .
09 m, k = 1200 N/m, ǫ = 6000 N/ m / and L = 0 .
15 m.The best fit indicates damping parameters α = 0 . δ = 8 N, A = 80 . - - - - - Λ real part Λ i m a g i n a r yp a r t - - - - - Λ real part Λ i m a g i n a r yp a r t Figure 7: Eigenvalues for the linearized system of differential equations for a set of pa-rameters ranging k = 5 ,
000 N/m, M = 2 . m = 0 . α = 0 − · − − − A = 0 − △ = δ = 0 N. 14 = 1 .
13 N.The most interesting result we noticed is that the nonlinear oscillations of the payloadinside the container, and the container motion are almost always in opposition of phase,that is one is maximum when the other is minimum. This effect seems to be independentof the initial conditions (drop height) the parameters of the container or payload, or thedissipation coefficients. We present such a study in Fig. 13 where we plotted contourplots of the acceleration of the container and the absolute acceleration of the payload onthe same ( t, k ) plane. The horizontal direction in these four plots is time line, and thevertical direction shows different values for the elastic constant of the payload connectingsprings, k . The two upper windows in Fig. 13 represent systems without dissipation, leftbeing dropped from low height, and right frame from high height. The bottom row offrames represent the same height (left and right) as above, except the system has regulardissipation, drag and friction as usual. WE note that the center lines of the contourlevels always coincides in the first oscillation. This is a strong argument in favor of thesynchronization of the two motions. In other words, we have a lock in of the phase ofthe two sub-systems, which guarantees in any situation (any drop height, any terminalvelocity, any initial energy, any dissipation parameters, etc.) the minimal acceleration ofthe payload, hence its protection. - - - - - - - - Λ real part Λ i m a g i n a r yp a r t Figure 8: Eigenvalues for the linearized system of differential equations for a set of pa-rameters ranging k = 10 N/m, M = 0 .
55 Kg, m = 0 . α = 0 − · − − − A = 0 − △ = δ = 0 N. 15 Study of the influence of the ground type
We experimented four types of drops on different grounds: concrete, sand, grass (forexample Fig. 14) and water (for example Fig. 15)There are differences in the behavior of the accelerations, but also similarities. In Fig.16 we present the time profiles of the energy for three types of ground. One can note adifferent rate of releasing the energy as well as phase shift in the oscillations. At the sametime, the inherent nature of the process remain independent of the type of ground.One can notice the same degree of synchronism of the maxima and minima, both inposition and acceleration for all types of impacts. We calculated the overlap between the @ s D Payload absolute position @ m D Calculated sphere positionSphere acceleration.Positive value means lift off tendency.Payload absoluteaccelerationMeasured sphere positions and errors - - t @ s D Sphere position @ m D , and acceleration @ - ms - D Figure 9: Theoretical (solid curves) and experimental (dots with error bars) results fora drop from H = 2 . m on concrete, see also Fig.2. The upper solid line represents tomotion of the center of mass of the container, beginning at the impact moment (labeled t = 0) and placed at the container radius height L , reconstructed from the rapid cameraimages, Fig. 2. The two lower curves represent the accelerations (solid for the container,and dashed for the payload) beginning at t = 0 , g = − . m/s . In the inset we presentthe result of the integration of the payload acceleration data in order to obtain the motionof the payload with respect to the ground. 16elocities profiles of the container and payload Z T payload t =0 Z ′ ( t ) · z ′ ( t ) dt, (17)on a period of the payload oscillation, which expression is actually a measurement of thedegree of asynchronism of the relative motion of the payload and the container motion.The results are presented in the Table 2.In Table 3 we present the maximum values of the payload absolute acceleration mea-sured in g’s for different types of ground. The values are taken from experiments, andverified for match with the numerical code results and errors less then 10%.Figure 10: Complete time line of the drop modeling presented in Fig. 9. In the upperpart of the frame we present the motion of the container center of mass (solid curve) andof the payload (dashed curve), both centered at heights L and L − δL , respectively.The gray stripe represents the motion of the upper and bottom wall of the container,and if the payload curve does not exit this stripe it means the payload did not collidedinto the container walls. In the lower part of the frame, we present the accelerationsof the container (solid curve) and payload (dashed curve) both centered at their initialacceleration − g . The coupled motion can be explained as the repetition of four mainphases labeled from I to IV and described in the text.17 ontainer accelerationPayload accelerationElastic energyPayload position Container positionKinetic energy - - - Figure 11: Time evolution of the energies involved in the impact in comparison withmotion and acceleration. In the inset we present a longer time scale which shows the rateof dissipation of energy (thick solid curve).
Total energy Kinetic energyElastic energyPayload accelerationPayload position - t @ s D Figure 12: Detail of the energy balance during bouncing off after impact.18
Conclusions
In this study we investigated the dynamics of the impact of a solid container and a payloadby dropping from different heights (from H = 0 . H = 10 m) on different types ofground (concrete, sand, grass and water). We ran experiments and measure instantaneousFigure 13: Contour plots of the acceleration of the container ( Z ′′ , dashed curves) andof the payload ( z ′′ + Z ′′ , solid curves) function of time, for different values of the elasticconstant k . In all the cases we choose M = 1 Kg, m = 0 . l = l = 0 .
09 m, R = 1 . l ,and L = 0 .
17 m. The differences between the figures are, clockwise from top left:
Upperleft= ǫ = 3500 N/ m / , H = 1 m and no dissipation or drag. Upper right= ǫ = 3500N/ m / , H = 10 m, and no dissipation or drag. Bottom left= ǫ = 6000 N/ m / , H = 1m, α = 0 . A = 80 Ns/m, δ = 8 . △ = 400 N. Bottom right= ǫ = 6000N/ m / , H = 10 m, α = 0 . A = 80 Ns/m, δ = 8 . △ = 400 N.19cceleration of both the container and the payload inside, and we taped the motion ofthe container using a rapid photography camera and take the image against a chessboardwall. We also elaborated a nonlinear one dimensional two-degrees of freedom model tosimulate the evolution and calculate accelerations and energies. The comparison betweenFigure 14: Impact on grass.Figure 15: Waves and surge created by impact on water.20he experiment and theory was very good. For the special system of nonlinear springswe designed we noticed a similar type of behavior of the payload for all types of groundin which the amplitude and acceleration of the payload is always in opposition of phasewith the ones of the container. This phenomenon reveals the possibility of the existenceof a energy resonant damping which allows a faster transfer of the shock energy from thepayload back to the container. This may be the signature of a non-linear energy sinkprocess and this topic will be studied in a forthcoming paper. References and Notes [1] O. A. Yakimenko, N. J. Slegers and R. A. Tiaden,
Development and Testing of theMiniature Aerial Delivery System Snowflake in the 20 th AIAA Aerodyn. DeceleratorSyst. Techn. Conf. and Seminar (4-7 May, 2009, Seattle, WA).[2] G. Kerschen, A. F. Vakakis, Y. S. Lee, D. M. McFarland, J. J. Kowtko and L. A.Bergman,
Nonlinear Dynamics (2005) 283-303.[3] A. F. Vakakis, L. I. Manevitch, Y. V. Mikhlin, V. N. Pilipchuk and A. A. Zevin Normal Modes and Localization in Nonlinear Systems (John Willey & Sins, Inc.,New York 1996).[4] A. Ludu and H. Liu,
Grant Report 1244967/SW01.1 (Embry-Riddle AeronauticalUniversity, June 2014).[5] J. P. Wilkinson,
J. Acoust. Soc. Am. (1965) 367-368. Concrete Sand Water
Figure 16: The energy dissipation profiles versus time for different types of ground.216] F. I. Niordson,
Int. J. Solids Struct. (1984) 667-687.[7] W. N. Findley, J. S. Lai and K. Onaran, Creep and Relaxation of Nonlinear Vis-coelastic Materials with an Introduction to Linear Viscoelasticity (North-Holland,Amsterdam, 1976).[8] T. A. Duffey, J. E. Pepin, A. N. Robertson, M. L. Steinzig and K. Coleman,
J. Vib.Acoust. , 3 (2007) 363-370.[9] A. E. H. Love,
A Treatise on the Mathematical Theory of Elasticity (Dover, NewYork 2011).[10] L. Pauchard, Y. Pomeau and S. Rica,
C. R. Acad. Sci. Paris , II b (1997) 411-418;L. Pauchard and S. Rica,
Phil. Magazine , 2 (1998) 225-233.[11] E. Efrati, E. Sharon and R. Kupferman, J. Meach. Phys. Solids (2009) 762-775.[12] Y. Klein, S. Venkataramani and E. Sharon, Phys. Rev. Lett. (2011) 118303.[13] N. Euler and M Euler,
J. Nonlinear Math. Phys. , 3 (2004) 399-421.[14] W. Nakpim and S. V. Meleshko, Symm. Integr. Geometry: Methods and Applic. ,051 (2010) 1-11.[15] C. Scheffczyk, U. Parlitz, T. Kurz, W. Knop and W. Lauterborn, Phys. Rev. A ,12 (1991) 6495-6502.[16] M. C. Nucci and P. G. L. Leach, J. Nonlin. Math. Phys. , 4 (2009) 431-441; M. C.Nucci and K. M. Tamizhmani, J. Nonlin. Math. Phys. , 2 (2010) 167-178.[17] M. C. Nucci and P. G. L. Leach, J. Math. Phys. , 1 (2007) 013514; G. D’Ambrosiand M. C. Nucci, J. Nonlin. Math. Phys. , Suppl. (2009) 61-71; M. C. Nucci andP. G. L. Leach, J. Nonlin. Math. Phys. , 4 (2009) 431-441; M. C. Nucci and K. M.Tamizhmani, J. Nonlin. Math Phys. , 3 (2012) 1250021.[18] Z. E. Musielak, J. Phys. A: Math. Theor. (2008) 055205; Z. E. Musielak, D. Royand L. D. Swift, Chaos, Solitons and Fractals (2008) 894-902.[19] L. Cveti´ c anin, Publ. L’Institute Math´ematique , 99 (2009) 119-130.[20] C. R. Galley, Int. J. Nonlin. Mech. , 2 (1987) 125-138.[21] J. Anderson, Fundamentals of Aerodynamics (McGraw-Hill, Columbus 2001).22able 1: Theoretical normal modes of oscillations (frequencies) of the container sphereused in experiments and simulations. i f ( i )[ Hz ] = - 22.1 26.3 28.1 29.2 f ( i )[ Hz ] = 65.3 89.4 118 150 181 f ( i )[ kHz ] = 2.93 2.94 2.94 2.95 2.96Table 2: The degree of synchronism between the two oscillations, Eq.(17), container andpayload, for different impact velocities.Ground type: Concrete Grass Sand Water V = 2 m/s 0.848 47.7 1.36 24.2 V = 10 m/s -14.3 49.3 2.21 29.9 V = 15 m/s -26.2 50.4 4.04 30.8Table 3: Maximum absolute acceleration max | Z ′′ ( t ) + z ′′ ( t ) | of the payload presented inunits of g . Ground type: Concrete Grass Sand Water V = 2 m/s 24.9 18.3 19.3 8.39 V = 10 m/s 105 77.0 54.2 61.3 V0