Generating Steady-State Chain Fountains
GG ENERATING S TEADY -S TATE C HAIN F OUNTAINS
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Johannes Mayet
Chair of Applied MechanicsDepartment of Mechanical EngineeringTechnical University of Munich [email protected]
Friedrich Pfeiffer ∗ Chair of Applied MechanicsDepartment of Mechanical EngineeringTechnical University of Munich [email protected] A BSTRACT
In recent years the chain fountain became prominent for its counter-intuitive fascinating phys-ical behavior. Most widely known is the experiment in which a long chain leaves an elevatedbeaker like a fountain and falls to the ground under the influence of gravity. The observed chainfountain was precisely described and predicted by an inverted catenary in several publications.The underlying assumptions are a stationary fountain and the knowledge of the boundary con-ditions, the ground and beaker reaction forces. In contrast to determining the steady-state chainfountain shape, it turns out that the main difficulty lies in predicting the reaction forces. A con-sistent and complete physical explanation model is currently not available. In order to give areasonable explanation for the reaction forces an illustrative mechanical system for generatingsteady-state chain fountain is proposed in this work. The model allows to generate all physicalpossible chain fountains by adjusting a pulley arrangement. The simplifications incorporatedmake the phenomenon accessible to undergraduate students.
Since 2013 when Steve Mould [
1, 2 ] presented its “self-siphoning beads” in a video, the chain fountain mysteryhas reached the public attention. Beginning with the 25th International Young Physicists’ Tournament in 2012 theproblem has been addressed by many scientists nowadays. In the chain fountain experiment one observes how along chain leaves an elevated beaker like a fountain and falls to ground under the influence of gravity (cf. Fig. 1).From a scientific point of view the underlying mechanical problem can be split into two separate problems. First,one has to determine an appropriate chain model, e.g. inextensible string or connected rigid bars, with which itis possible to represent chain properties like dissipation and bending stiffness in a quantitative manner in orderto address specific phenomena, e.g. traveling waves. Second, one faces the challenge to describe the (averaged)dynamics of folding and unfolding chains because it needs a mechanism to accelerate the chain links from theirresting state to constant speed and vice versa. These problems are very well known since a long time, see forexample Routh [ ] or Hamel [ ] , who consider continuous chain models. Certainly, publications concerning theparadoxical phenomenon that the free end of a vertically hanging folded chain is falling faster than a free-fallingbody under gravitational acceleration have also to be considered. This topic is addressed by Schagerl [ ] andSteiner [ ] in experiments and numerical simulations, for example. In reference [ ] it is shown that the geom-etry of the folding at the bottom is crucial and has to be accounted for in the dynamics of the chain. The fallingchain experiments indicate that a ball chain with a minimal radius of curvature is more accelerated in contrast toclassical loop chains with no well-defined minimum radius of curvature (cf. [ ] ). Grewal [ ] suggests differentmechanical designs to increase the phenomenon of a down-pulling falling chain. Willerding [ ] considers spaceproblems connected with chains and ropes, but also the falling U-shape chain and the chain fountain.At the International Young Physicists Tournament ( IYPT ) in 2012 several interesting measurements gave attentionto unfolding mechanisms of the chain. Hledik [ ] and Santiago [ ] show results with initial chain arrangementswhich are well-ordered (see Fig. 1c) or chaotic (see Fig. 1a) yielding completely different fountains, as illus-trated in Fig. 1d and Fig. 1b. This result has been recently confirmed by Martins [ ] . These observations are ∗ a r X i v : . [ phy s i c s . c l a ss - ph ] A ug enerating Steady-State Chain Fountains A P
REPRINT (a) Initial configuration. (b) Fountain of (a). (c) Initial configuration. (d) Fountain of (c).
Figure 1: The random initial setting shown in (a) yields a much higher fountain depicted in (b) compared tothe initial setting (c) and corresponding fountain (d) when falling from the same height (pictures are takenfrom [ ] ).conceptually in agreement with the unfolding process as dissipationless extraction of chains on a flat table, cf.reference [
19, 8 ] . Flekkøy, Moura and Måløy [ ] study based on simulations different types of chains as wellas surface conditions of the beaker. On the one hand, the importance of a rough beaker bottom is emphasizedsince it causes a different unfolding behavior of the chain. On the other hand, the relevance of different chaintypes, which are modeled by different internal stiffness interactions, is pointed out. Virga [ ] , however, usesa continuous chain model and applies "dissipative shocks" at the beaker and floor by arguing that the pick-upand put-down processes take place in a very short time. Interestingly collisions are a key concept in Grewal’s [ ] designs increasing the falling chain phenomenon. Although shock propagation and internal stiffness interactionsare completely different approaches, both theories yield convincing results with respect to characteristic fountainparameters.Biggins and Warner [ ] derive an inverted caternary fountain by using a simple inextensible string model. Theinverted caternary is in good agreement with their experimental results and those of Pantalenone’s [ ] quantita-tive study. The inverted caternary fountain, however, requires reaction forces on the starting point and endpointof the fountain. The associated reaction forces are interpreted in reference [ ] by collisions of each chain elementwith the support when lifted up. Despite the fact that the influence of different initial chain arrangements cannot be explained reasonable, this approach is quite controversial due to high-speed camera videos showing thatthe chain elements can leave their support in a horizontal direction for a long time without affecting the overallchain fountain.In contrast to the above-mentioned publications the aim of this work is to present a simple principle model ofthe chain fountain for which it is possible to derive the steady-state behavior by fundamental mechanical laws.Speculative forces arising from the table and floor will not be required since the beaker and the floor interactionsare replaced by a pulley arrangement allowing for a continuous acceleration and deceleration of the chain ele-ments. As a consequence, it is not necessary to deal with shock propagation and traveling waves. However, theenormous advantage of this principle model is also the biggest disadvantage, since the complex system behaviorcan only be approximated. No conclusions can be drawn about the real folding and unfolding mechanism.The paper is organized as follows: In the first part the inverted caternary is derived by assuming apriori knownbeaker and floor interaction forces yielding the free flight dynamics of the chain. The following section discussesthe principle model for folding and unfolding the chain at the end points, which allows the explanation of reactionforces within their limits. Finally, the overall principle model of the chain model is presented and the underlyingequations for a steady-state fountain are given. 2enerating Steady-State Chain Fountains A P
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In this section the equations of motion for a steady-state chain from Pfeiffer [ ] are recapitulated. In contrast tothe equations of motion for thin strings the equations are extended by additional rotational stiffness and dampingeffects. However, it is assumed that the chain is inextensible ( || (cid:126) x (cid:48) || = () (cid:48) = ∂ /∂ s is the partial derivativew.r.t. the arc-length s of the fountain), the density is constant and more importantly the velocity is constant( || ˙ (cid:126) x || = v ). A useful choice for the representation of the velocity is then given by ˙ (cid:126) x = v (cid:2) cos ( α ) sin ( α ) (cid:3) T sincethe assumptions || ˙ (cid:126) x || = v and || (cid:126) x (cid:48) || = O V E hHl (a) Chain fountain parameters. T ( s + ∆ s ) T ( s ) ∆ s ∆ m a I x I y ˙ x α ∆ m = ρA ∆ sQ ( s + ∆ s ) Q ( s ) M ( s + ∆ s ) M ( s ) (b) Infinitesimal belt segment and cutting forces. Figure 2: The chain fountain is described by its overshoot height h , falling down height H and horizontal distance l between the beaker and the approximated landing area on the floor (left). Internal forces in tangential andnormal direction as well as a bending torque act on a gravitational attracted infinitesimal belt segment of mass ∆ m and length ∆ s (right).chain element yields¨ (cid:126) x ( s , t ) λ ( s ) ∆ s = − (cid:126) T ( s , t ) + (cid:126) T ( s + ∆ s , t ) − (cid:126) Q ( s , t ) + (cid:126) Q ( s + ∆ s , t ) + (cid:126) a ( s , t ) λ ( s ) ∆ s ,where λ ( s ) is the linear mass density ρ ( s ) times cross-sectional area A , (cid:126) T and (cid:126) Q are vectors of internal forces and (cid:126) a ( s , t ) is the vector of gravitational acceleration acting on the segment with length ∆ s , see Fig. 2b. The curve ofthe chain is described by (cid:126) x = (cid:2) x p y p (cid:3) T with the choice x p = y p = ∆ s → λ ( s ) ¨ (cid:126) x ( s , t ) = ∂∂ s (cid:126) T ( s , t ) + ∂∂ s (cid:126) Q ( s , t ) + λ ( s ) (cid:126) a ( s , t ) . (1)The internal cutting forces are chosen such that (cid:126) T = T ( s , t ) (cid:2) cos ( α ) sin ( α ) (cid:3) T acts in direction of the tangentvector and (cid:126) Q = Q ( s , t ) (cid:2) sin ( α ) − cos ( α ) (cid:3) T acts in direction of the normal vector. Applying the steady-stateassumptions (˙ s = v ) the equations of motion can be written as α (cid:48) (cid:0) ¯ T − (cid:1) + ¯ Q (cid:48) = k cos α and ¯ T (cid:48) − α (cid:48) ¯ Q = k sin α , (2)where the scaled internal forces ¯ T = T / ( λ v ) and ¯ Q = Q / ( λ v ) , and the constant k = g / v accounting forgravitational forces ( (cid:126) a = [ − g ] T ) have been introduced. Usually the internal forces ¯ Q are neglected formaterials like ropes and strings since one assumes zero bending stiffness for such materials. In the case of theconsidered chain, which consists basically of small rigid rods connected by spherical joints, normal forces ¯ Q have actually to be considered even if the chain elements are assumed to be (infinitesimal) small. If a rotationalspring (coefficient c ) and damper (coefficient d ) resistance of the spherical joints are introduced, then the angularmomentum equation with mass moment of inertia J λ ∆ s is J λ ∆ s ¨ α ( s , t ) = − ∆ s ( Q ( s + ∆ s , t ) + Q ( s , t )) − c ( α ( s + ∆ s , t ) − α ( s , t )) − d ( ˙ α ( s + ∆ s , t ) − ˙ α ( s , t )) ,3enerating Steady-State Chain Fountains A P
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12 0 .
14 0 .
16 0 .
18 0 . κ = l/H ¯ q [ − ] ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 .
20 0 .
06 0 .
08 0 . .
12 0 .
14 0 .
16 0 .
18 0 . ξ = h/H ¯ q [ − ] κ = 0 . κ = 0 . κ = 0 . κ = 0 . Figure 3: Numerical solution of ¯ q = Hq using Eq. (5) for given values of ξ = h / H and κ = l / H .and by approaching the limit ∆ s → Q = − J α (cid:48)(cid:48) − c / ( λ v ) α (cid:48) − d / ( λ v ) α (cid:48)(cid:48) . Further, if a spherical jointresistance, which linearly depends on the tension along the curve, is assumed, one may postulate a mathematicalmotivated ansatz ¯ Q = − (cid:0) (cid:34) + (cid:34) ¯ T (cid:1) α (cid:48) − (cid:0) (cid:34) + (cid:34) ¯ T (cid:1) α (cid:48)(cid:48) with all (cid:34) i ∈ (cid:82) + (cid:28) (cid:34) i are only addressed numerically from a phenomenologicalpoint, because experiments that reveal physical parameters are not available at the present time. In order tounderstand the underlying physical relationships of the chain fountain, the solution curves for all (cid:34) i = In the case that all (cid:34) i are equal zero (cf. Biggins [ ] ), the resulting differential equations α (cid:48) (cid:0) ¯ T − (cid:1) = k cos α and¯ T (cid:48) = k sin α lead directly to the solution for ¯ T = − k / ( q cos α ) in terms of α ( s ) and one remaining differentialequation α (cid:48) = q cos α with q = k (cid:2) ( − ¯ T E ) cos α E (cid:3) − >
0. The constant parameter q takes into account thatthe tension at endpoint, given by α ( s = s E ) = α E ∈ [ π/ ] , is equal to ¯ T E ∈ [
0, 1 [ . Although determining asolution for α = α ( s ) means no serious difficulties, it is advantageous to determine a solution for y p = y p ( α ) using y (cid:48) p = sin α in order to establish the relation to the geometrical dimensions of the fountain:d y p = sin α d s = sin α d s d α d α → cos α = / ( − q y p ) (3)Using d y p = tan α d x p we obtain the solution curve as function graph q y p = − cosh (cid:0) qx p (cid:1) . (4)This inverted caternary coincides with the results of Biggins [ ] . This solution representation stands for itssimplicity. Further, a severe conclusion can be drawn: As it is possible to scale lengths by q , e.g. ¯ y p = q y p and ¯ x p = qx p , yielding a parameter independent solution curve, it is obvious that one can not state an additionalmechanical law for the free flight dynamics which can be used to determine the unknown parameter q respectivelythe unknown reaction forces. Note that this actually means that the constants q and k arising in the cutting force¯ T = − k / q + k y p < T and T E occur at the endpoints and are assumed to be constantdue to the steady-state assumption. But questions on how these cutting forces really originate can not be answeredsince an overall system model would be required.Employing again the function graph representation given in Eq. (4) we derive the relationship cos α = (cid:0) − q y p (cid:1) − which is used to obtain an alternative formulation for the endpoint angles by taking the beaker location at height y p = − h and the floor height y p = − h − H into account, see Fig. 2b: ql = acosh ( + q ( h + H )) + acosh ( + qh ) (5)Consequently, the parameter q is implicitly given by Eq. (5) and therefore the fountain is completely specified bythree points (beaker, floor and vertex), respectively the parameters l , h and H . In Fig. 3 numerical solutions ofEq. (5) for ¯ q = Hq are depicted. It is important to note that the shape of the fountain can not be altered in the4enerating Steady-State Chain Fountains A P
REPRINT case of the proposed mechanical model. A physical interpretation of the parameter q is revealed by evaluatingthe signed curvature ( x (cid:48) y (cid:48)(cid:48) − x (cid:48)(cid:48) y (cid:48) ) / ( x (cid:48) + y (cid:48) ) / = α (cid:48) = q cos α at the vertex with α =
0. This result is by farnot intuitive since it might be expected for example that fountains can be realized having different shapes due todifferent traveling velocities.Since q is already defined by kinematic relationships the remaining unknown parameter k = g / v has to bedetermined by measuring the average velocity of the chain elements. Although measuring the average velocitydoes not present any particular difficulties, it useful to restrict the possible parameter range for k by assumingtensile forces: ¯ T = − kq − kh ≥ T E = − kq − k ( h + H ) ≥ → k ≤ q + ( h + H ) q < h + H (6b)Therefore, it can be concluded using the above inequality for k that the average velocity v has to be greater than v min = (cid:112) g ( h + H ) . The velocity v min is equal to the vertical velocity of a rigid body at ground when dropped atheight h + H with zero initial vertical velocity. In the case of ¯ T E ≈ q (cid:29) v and theminimal velocity v min are indeed approximately equal. In order to obtain a lower bound for the parameter k itis assumed that conservation of energy for the free flight chain segment holds, which yields v max = (cid:112) gH . Theresulting lower bound k > / ( H ) and as a result qH > / ( − ξ ) > ξ = h / H ∈ [
0, 1 [ are very roughestimates at best. But without measuring the average velocity v it is now possible to restrict the end-point forces:0 ≤ ¯ T E < − (cid:129) qH + + ξ (cid:139) <
12 (7a)12 < qH + qH ( + ξ ) ≤ ¯ T < − kq < − qH < ( + ξ ) < T E ∈ S E ⊆ [
0, 1 / [ and ¯ T ∈ S ⊆ ] /
2, 1 [ has to be satisfied for allchain fountains using a continuous string model. However, as previously explained it is not possible to identifyunique values for ¯ T E and ¯ T without measuring the average velocity v of the chain. In Fig. 4 an example fora chain fountain (fountain D ) is depicted with two different values for k ( v = [ m / s ] and v = [ m / s ] )as it may be observed in the experiment. As expected, the fountain cannot be distinguished since the chainparameter l , h and H are the same in both cases. However, in the case of k max = q / ( + ( + ξ ) qH ) the groundreaction force ¯ T E is equal zero in contrast to the case where the chain moves 0.3 [ m / s ] slower. As illustratedin Fig. 4, it must be kept in mind that these statements are only permissible under the assumption of a stringmodel. The additional internal force ¯ Q which is motivated by the spherical joint resistance and mass moment ofinertia gives rise to chain fountains with possibly higher overshoot heights h . Different heights do not necessarilycause significant differences in horizontal displacement l as depicted by fountains A and B . Conversely, differenthorizontal displacements can be generated whereby maintaining the overshoot height.As an intermediate conclusion it can be said that the presented chain model approximates the complex dynamicalbehaviour reasonably well. It is essentially useful when ruling out non-physical parameters or reaction forces,but discrepancies will inevitably remain. The question which naturally arises is how one can obtain the specified ranges for the forces given in Eq. (7) bypicking up chain elements. Commonly one is tempted to average the dynamical behavior by applying the lawof conservation of momentum for a particle, which is at rest and afterwards it travels with velocity v within aninfinitesimal short time, leading to (cid:90) t + t − T dt = ∆ m ( v ( t + ) − v ( t − )) → ¯ T = T has to be smaller than the maximalforce T max = − k / q <
1. Since the chain is treated as a chain of particles this approach approximates the pick-upprocess well if the chain is made of single elements connected by strings. In our case, however, the chain has thedistinctive property of a minimal radius of curvature with the result that the chain is unfolded rather than thatindividual chain particles are accelerated by an impact. Therefore, we follow in this work a simple contemplation5enerating Steady-State Chain Fountains
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REPRINT . . . . − − . − . − . − . − − . − . − . − . . x [m] y [ m ] ABCD . . . . . . . . x [m] ¯ T ABCD : k = 0 . D : k = k max = 0 . . . . . − π − π − π π x [m] α [ r a d ] ABCD
Figure 4: Example for results obtained for a chain fountain using the belt approximation: left side the fountain,right side top the scaled internal chain tension, and right side bottom the fountain angle α for different parameters.Equal parameters: H = [ m ] , ξ = h / H = κ = l / H = Hq = (cid:34) = (cid:34) =
0. Fountain A: (cid:34) = e − (cid:34) = e − k = (cid:34) = e − (cid:34) = e − k = (cid:34) = e − (cid:34) = e − k = (cid:34) = (cid:34) = k = k = T does not necessarily has to be close to 1. As illustrated in Fig. 5 the pick-upprocess is described by chain with zero bending stiffness guided by a massless roller. As depicted on the left sidein Fig. 5 a pulling force F = λ v / ( + cos α ) is required to reach steady-state. This result is obtained by notingthat the velocity of the point P is given by ˙ (cid:126) x P = (cid:2) v R (cid:3) T − r ˙ α (cid:2) cos α sin α (cid:3) T with roller radius r . The angularvelocity is given by ˙ α = − v R / r , because the lower part of the chain is not moving. The velocity of point P alongthe inclined direction is then (cid:2) cos α sin α (cid:3) ˙ (cid:126) x P = v R ( + cos α ) . Consequently, we have v R = v / ( + cos α ) , whichis the translational velocity of the roller and therefore the mass increase velocity. Conservation of momentumthen requires a pulling force F = λ v / ( + cos α ) . In a first limit case consideration, it can be concluded thatthe true scaled force probably originates from a mechanism that is something between this simplified unfoldingmechanism with α ∈ [ π/ [ and a pure impulse that is equivalent to α = π/
2. On the other hand, it is alsopossible to estimate the pull-down forces acting on the chain with the same basic consideration. Assuming that thepull-down process is an average of chain elements hitting directly the table with F = A P
REPRINT v/ v F vv/ v/ v/ v FFF v v R vv v l h = ξ Hξ H ξ H vv v/ v/ − ξ ) H ¯ T = + ¯ T lc ¯ T ¯ T ¯ T E ¯ T = + ¯ T lc − ξ kH ¯ T ¯ T = + ¯ T uc + ξ kH ¯ T = + ¯ T uc ¯ T ¯ T vv v/ v/ α P Figure 5: Schematic steady-state unfolding and folding process: The limited curvature of the chain is taken intoaccount by the rollers. Left: As the upper part of the chain travels with velocity v the mass increase is dm = λ v R d t with v R = v / ( + cos ( α )) . As a consequence, the pulling force F has to be F = λ v / ( + cos ( α )) . Right: As theupper part of the chain travels with velocity − v the mass decrease is dm = λ v d t . As a result the pull-down force F is given by F = λ v .e.g. the phenomenon of the U-shape or the “sucking” chain [
5, 15, 16 ] . These observations permit now theconstruction of an apparatus with which steady-state chain fountains can be generated. In order to be able to generate steady-state chain fountains we use the above presented concept of guiding pulleys.In Fig. 6 an example for an arrangement of the pulleys is illustrated. Due to the pulleys it is possible to continuouslyaccelerate and decelerate the chain elements. Since the radius of the roller is irrelevant for the pulling force, itis assumed that the radius is sufficiently small and the influence on the force balance can be neglected for thesake of simplicity. Furthermore, the principle model should be considered idealized in terms of non-conservativeinfluences such as dissipation and friction. As depicted in Fig. 7 it is possible to obtain reaction forces ¯ T E = + ¯ T lc − ξ kH and ¯ T = + ¯ T uc + ξ kH where ¯ T lc > T uc > T − ¯ T E = kH has tobe satisfied. This can be achieved by setting ¯ T uc = ¯ T lc + kH ( − ξ − ξ ) and as a result ¯ T = + ¯ T lc + kH ( − ξ ) .Certainly, this requirement is equivalent to conservation of energy which is consistent with our assumptions.The enormous advantage of this approach is that the chain fountain can be determined in a two-step procedure.First the fountain kinematics are defined and the associated forces of the free chain are calculated. In the nextstep, the remaining heights and the cart-masses would be chosen in such a way that the necessary forces areachieved. It should be kept in mind that it is theoretically possible to generate all relevant reaction forces if thefree flight dynamics of the chain are captured accordingly with the proposed simple string model. Taking intoaccount the already derived value ranges of the reaction forces one obtains ξ kH − ¯ T lc ≤ − kH . One special casethat comes to the surface is ξ = ξ = /
2, because the energy conservation is immediately visible. The potentialenergy of the chain elements in the beaker and those on the bottom are exactly the same at any given time andthe carts balance each other out. Furthermore, the number of chain elements that are in motion stays the same.Consequently, it is expected that the existence of this case eliminates the possibility to describe the phenomenonwith potential energy transfer. Due to the fact that ¯ T = + ¯ T lc + kH / T = − kq − kh , the specialcase ξ = ξ = / h , H and q with the parameter k if ¯ T lc = k = k max = q / ( + q ( h + H )) forexample is obtained for ξ = / ( kH ) and ¯ T lc =
0. It yields ¯ T = kH and as a result ¯ T E =
0. Since the proposedsubstitute model supposedly does not precisely reflect all relationships of the chain fountain, as an approximationto the original experiment, one can omit the lower part of the arrangement and simply drop the chain on the flooras in the true experiment, see Fig. 8. In any case, it is advantageous that the complex dynamic processes duringcollision of chain elements with the bottom can be studied separately to the pickup process. Experiments withnon-stationary fountains could provide information about the material behaviour, e.g. it is certainly advantageousto observe the time-dependent system behaviour changes to quantify dissipative influences. As indicated in Fig. 8,an arrangement can be specified in a simple manner in which the reaction force change is exactly known overtime. 7enerating Steady-State Chain Fountains
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REPRINT v/ v F vv/ v/ v/ v FFF v v R vv v l h = ξ Hξ H ξ H vv v/ v/ − ξ ) H ¯ T = + ¯ T lc ¯ T ¯ T ¯ T E ¯ T = + ¯ T lc − ξ kH ¯ T ¯ T = + ¯ T uc + ξ kH ¯ T = + ¯ T uc ¯ T ¯ T vv v/ v/ α P Figure 6: An arrangement of pulleys to decelerate (left) and accelerate (upper right) the chain elements fromrest to constant velocity v and vice versa in order to generate a steady-state chain fountain (blue). The additionalcarts on an inclined plane allow to manipulate the tension in order to get different values than ¯ T = /
2, seealso Fig. 5. Positioning the "bottom” by ξ H above the chain endpoint and the “beaker” ξ H under the startingpoint of the chain gives the opportunity to make further adjustments. It should be emphasized that the rollersare displayed over-proportionally. As a summary, the derivation of the chain fountain with simplified spherical joint resistance torque is presented.It was shown that identical fountains can be generated with completely different force characteristics. The key tothis is the insight that already the kinematic parameters l , h and H are completely defining the fountain. Sincethe free chain is just cut out of the overall system, the kinetic parameters cannot be uniquely determined withsuch a subsystem consideration. As a consequence, only value ranges can be specified. The core of this work,however, is to find a simple explanation for the necessary cutting forces. The presented substitute system for theunfolding and folding process allows the whole range of values for the reaction forces to be covered. Even if it isnot possible to entirely explain the chain fountain in the cited videos with the proposed mechanism, it has beenshown that a steady-state chain fountain can be generated without additional argumentation such as travelingwaves, dissipation effects, shock propagation, transverse motions or ground kicks. References [ ] The chain fountain aka the mould effect. . Ac-cessed: 2019-12-09. [ ] Investigating the "mould effect" | steve mould | tedxnewcastle. . Accessed: 2019-12-09. [ ] John S Biggins and Mark Warner. Understanding the chain fountain.
Proceedings of the Royal Society A:Mathematical, Physical and Engineering Sciences , 470(2163):20130689, 2014. [ ] Eirik Flekkøy, Marcel Moura, and Knut Måløy. Dynamics of the fluctuating flying chain.
Frontiers in Physics ,7:187, 11 2019. [ ] Anoop Grewal, Phillip Johnson, and Andy Ruina. A chain that speeds up, rather than slows, due to collisions:How compression can cause tension.
American Journal of Physics , 79(7):723–729, 2011.8enerating Steady-State Chain Fountains
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REPRINT v/ v F vv/ v/ v v/ v/ v F FFv vvv v l h = ξ Hξ H ξ H vv v/ v/ − ξ ) H ¯ T = + ¯ T lc ¯ T ¯ T ¯ T E ¯ T = + ¯ T lc − ξ kH ¯ T ¯ T = + ¯ T uc + ξ kH ¯ T = + ¯ T uc ¯ T ¯ T Figure 7: At crucial transition points the resulting reaction forces / tensions are illustrated. Additionally to thefolding, respectively unfolding mechanism from which one gets ¯ T = /
2, the carts are accounted for by ¯ T lc and¯ T uc , which are scaled quantities. The “floor” was raised in relation to the chain endpoint in order to be ableto generate values for ¯ T E smaller than 1 /
2. The “beaker” was lowered in relation to the starting point of thefountain even though the cart has the same influence. In both cases, the effect on the tension due to gravity wasconsidered. The influence of the rollers on the balance of forces was neglected for the sake of simplicity. v/ v F vv/ v/ v/ v FFF v vvv v l h = ξ Hξ H ξ H vv v/ v/ (1 − ξ ) H ¯ T = + ¯ T lc ¯ T ¯ T ¯ T E ¯ T = + ¯ T lc − ξ kH ¯ T ¯ T = + ¯ T uc + ξ kH ¯ T = + ¯ T uc ¯ T ¯ T vv v/ v/ Figure 8: Simplified pulley arrangements. Chain fountain with idealized beaker but real bottom (left). Furthersimplification of Fig. 6, but without subsequent deceleration of chain elements as more and more chain elementsare in motion over time (right). Experiments with non-stationary fountains are straightforward to conduct by notguiding the lower part of the chain horizontally. Different steady-state chain velocities can be easily achieved bydifferent lower cart masses. [ ] Georg Hamel. Theoretische Mechanik. Grundlehren der Mathematischen Wissenschaften, LVII.
Springer,Berlin , 100, 1949. [ ] Eugenio Hamm and Jean-Christophe Géminard. The weight of a falling chain, revisited.
American Journalof Physics , 78(8):828–833, 2010. [ ] J. A. Hanna and C. D. Santangelo. Slack dynamics on an unfurling string.
Phys. Rev. Lett. , 109:134301, Sep2012. [ ] Michal Hledík and Tomáš Bzdušek. String of beads. https://archive.iypt.org/solutions/ , 2012.International Young Physicists Tournament (IYPT).9enerating Steady-State Chain Fountains
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REPRINT [ ] Rogério Martins. The (not so simple!) chain fountain.
Experimental Mathematics , 28(4):398–403, 2019. [ ] J. Pantaleone. A quantitative analysis of the chain fountain.
American Journal of Physics , 85(6):414–421,2017. [ ] F. Pfeiffer and J. Mayet. Stationary dynamics of a chain fountain.
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