Nonlinear dynamics of a hanging string with a freely pivoting attached mass
Filip Novkoski Jules Fillette Chi-Tuong Pham Eric Falcon








1]Filip Novkoski

1,2]Jules Fillette
3]Chi-Tuong Pham

1]Eric Falconcor1
eric.falcon@u-paris.fr


[1]organization=Université Paris Cité, CNRS, MSC, UMR 7057, city=Paris, postcode=F-75013, country=France
[2]organization=LPENS, ENS, CNRS, UMR 8023, PSL Research University, Sorbonne Université, postcode=F-75005,city=Paris,country=France
[3]organization=Université Paris-Saclay, CNRS, LISN, UMR 9015, city=Orsay, postcode=F-91405, country=France

[cor1]Corresponding author




We show that the natural resonant frequency of a suspended flexible string is
  significantly modified (by one order of magnitude) by adding a freely pivoting attached mass at
  its lower end. This articulated system then exhibits complex nonlinear dynamics such as bending
  oscillations, similar to those of a swing becoming slack, thereby strongly modifying the system
  resonance that is found to be controlled by the length of the pivoting mass. The dynamics is
  experimentally studied using a remote and noninvasive magnetic parametric forcing. To do so, a
  permanent magnet is suspended by a flexible string above a vertically oscillating conductive
  plate. Harmonic and period-doubling instabilities are experimentally reported and are modeled
  using the Hill equation, leading to analytical solutions that accurately describe the
  experimentally observed tonguelike instability curves.

















Experiments; Articulated flexible pendulum; Parametric forcing; Nonlinear dynamics; Period doubling 


















§ INTRODUCTION

The vibrational modes of a flexible and inextensible hanging string in the gravitational field is a
classical mechanical problem, and its solution dates back to Bernoulli and Euler in the 18th century
<cit.>. The case of the flexible string with an additional mass at its lower end has
been considered only recently, both theoretically <cit.> and
experimentally <cit.>, and its modes are found to be close to those of a rigid
pendulum <cit.>. Although such a mechanical system involves a spatially dependent
  wave velocity (as for acoustics in air ducts <cit.> or in granular
  media <cit.>), there is little experimental data for such articulated pendulum systems
  compared to those involving rigid limbs such as the double pendulum <cit.>.

Here, we study a system constituted of a flexible string with a freely pivoting attached mass to
  its lower end. We show that its natural resonant frequency is one order of magnitude higher than
  the one of a single flexible string. Such a two-component articulated system exhibits bending
oscillations similar to those of a swing becoming slack, thus strongly modifying its
resonances. They are studied using an original technique based on noninvasive and remote magnetic
forcing. More precisely, we investigate the dynamics of a permanent magnet, freely pivoting,
suspended by a flexible string above a vertically oscillating conductive plate. Induced currents due
to the moving conductor generate remote forces on the magnet, giving rise to a parametric forcing
needed to study the nonlinear dynamics of the system. The main advantage of this forcing is to
  avoid the excitation of the string torsional modes and the single pendulum mode unlike vertically
  oscillating the top attachment point of the string. Note that parametric
instabilities <cit.>, phase transitions <cit.>, bifurcations <cit.>
and control of chaos <cit.> have been investigated with a rigid pendulum with a
magnet as their tip mass in an oscillating magnetic field <cit.>, or with a
ferrofluid drop suspended by a torsion pendulum <cit.>. For a flexible string,
with no tip mass and vertically vibrated, parametric resonance, chaos, and self-knotting have
been experimentally evidenced <cit.>.




























§ EXPERIMENTAL SETUP

The setup is shown in Fig. <ref>. It consists of a flexible nylon cord suspended at its
upper point and with a tip mass attached at its lower end using glue. The cord has a 0.35-mm
diameter, linear mass density ρ=1.24 mg/cm, and length L=6 cm. The tip mass consists of two
neodymium permanent magnets (square cuboid of size a× a×a/2, with a=1 cm,
total magnetic moment m=0.8±0.05 A/m^2) in which a 3D-printed thin plastic square
plate is sandwiched with a length of ℓ∈[0,2] cm protruding above the magnets (see
Fig. <ref>). The total tip mass is M=8 g.  The articulated system is thus constituted
  of a flexible string (of length L) with a rigid pendulum (of length ℓ+a) attached to its
  end. We force the system parametrically, using a noninvasive and remote magnetic forcing, to study
  its dynamics. To do so, the system is suspended above the center of a circular copper plate. The
plate is attached to an electromagnetic shaker using a long aluminum shaft to avoid any
interference from the internal magnetic field of the shaker. The plate is subjected to vertical
oscillations, Asin(2π f t), at a mean distance d_0∈[7,9] mm from the magnet center, with
f∈[15,40] Hz and A∈[0.07,2] mm, the frequency and amplitude of the vibrating plate,
  respectively. The relative distance between the magnets and the plate is
d(t)=d_0+Asin(2π f t). The horizontal velocity of the magnet is measured either by a Polytech laser vibrometer
for small amplitude motions or by a homemade copper wire coil, placed close to the magnet, for large
ones. The magnet motion is visualized by a Basler camera (2048×1536 px^2, 120 fps) located
in front of it.





 

§ FREQUENCY RESPONSE

 We first determine the frequency response of the magnet-string system. To do so, the forcing frequency f is varied
 linearly in time, during 2 min, from 15 to 50 Hz, and the horizontal velocity v(t) of the
 tip mass is recorded. We compute its frequency spectrum
 S_v(ω)∼ |v̂(ω)| [where v̂(ω) is the Fourier transform of v(t)]
 as shown in Fig. <ref>. The spectrum maxima correspond to the system frequencies
 f_r in response to different forcing frequencies f. We clearly see that the response follows
 the forcing, f_r=f, except in a region (see vertical dash-dotted lines), where a subharmonic
 behavior, f_r=f/2, is observed. In this range, the magnet exhibits large-amplitude oscillations
 (in the planar pendulum motion) in which the string swings back and forth at half the driving
 frequency, corresponding thus to a period-doubling instability [see the red region and photos in Fig. <ref>, and a movie in the Supplemental Material (see <ref>)]. The system
 exhibits also small rotations of the center of mass, around the equilibrium, with deviations
 of the string from the vertical. Such motion is restricted to the vertical y-z plane and no
 rotation around the vertical axis is found.  Outside the red region in Fig. <ref>, a low-amplitude motion is
 observed. Interestingly, the magnetic damping of the conductive plate cancels any torsional
 motion. 
 
 We now measure the natural resonant frequency f_0 of the magnet-string system. We move laterally
 the magnet away (over the y-axis) from its equilibrium with a small string flexion to excite only
 the mode observed in the inset of Fig. <ref> and not the unflexible pendulum one
 [occurring at a much lower frequency √(g/L)/(2π) ≃ 2 Hz]. We record the magnet
 velocity during its free-damped oscillations. Through a Fourier transform, the resonant frequency
 is inferred from the spectrum peak frequency, f_0, and is noted by a vertical dashed line in
 Fig. <ref>. The subharmonic region in Fig. <ref> occurs around
 2f_0, as expected for parametric resonance <cit.>, with the corresponding response
 f_r=f/2, i.e., a period-doubling motion. Figure <ref> also shows that no
 subharmonic instability occurs around f_0, as expected, whereas a second-harmonic instability,
 i.e., f_r=2f, is observed.


 


§ INSTABILITY THRESHOLD

The threshold of the subharmonic instability depends on the vibrating parameters f and A. It is
experimentally measured by slowly increasing A, for a given f, until observing an exponential
rise in the system oscillation amplitude. The experimental data are shown in Fig. <ref> to
collapse onto continuous curves in the nondimensional parameter space (A/d_0, f/f_0) which are
suggestive of two resonant curves around f_0 (in blue) and 2f_0 (in red). In the zone above the
red points, a period doubling of the oscillations of the bending string is observed, i.e.,
f_r=f/2, corresponding to the inset of Fig. <ref>.  The tonguelike shape of this
marginality curve is reminiscent of parametric resonance. The instability is also found slightly
hysteretic. In the region above the blue points, the system responds, as f_r=f, corresponding to a
harmonic instability.  Note that, for frequencies slightly close and above f_0, no experimental
data are indicated since we do not observe a clear transition to the harmonic instability (see
below).



§ BENDING STRING DESCRIPTION

The behavior of a hanging flexible string
 differs significantly from that of a rigid pendulum <cit.>. Let us consider a
 flexible and inextensible string suspended from its fixed upper point, subjected to gravity and
 tension forces, and a maximal displacement of its end X_0 at an initial time. The string position
 𝐗(s,t) is then described by Newton's law as <cit.>

    ρ∂_tt𝐗=∂_s[T(s,t)ŝ]-ρ gẑ

where s∈(0,L) is the curvilinear position along the string taking its origin at its upper end
[𝐗(0,t)=0], T(s) is the tension acting in the string, ρ is the linear
mass density, and g=9.81 m/s^2. By applying additional boundary conditions (BCs), T(L,t)=0 (no
tension acts on the free end), and 𝐗(L,0)=𝐗_0 (initial condition), the solutions
of Eq. (<ref>), for no tip mass and small displacements from the equilibrium configuration,
are found as modes of the zeroth-order Bessel function of the first kind
J_0 <cit.>. Nevertheless, due to the added tip mass, our system is able to rotate
around its center of mass (in the planar pendulum motion), and a significant modification of
its resonant frequencies is predicted <cit.>. Beyond BCs on the string position at its
ends, forces, and torques on the tip mass need to be theoretically considered leading to nontrivial
BCs <cit.>. In that case, solutions of Eq. (<ref>), for small displacements,
are that of a classical harmonic oscillator of frequencies ω_n, each corresponding to a
system mode. Note that if the string were elastic, one would recover a parametrically
  excited buckling beam, most likely governed by a Duffing-like equation <cit.>.

By considering small motions around equilibrium, Eq. (<ref>) reads

    ρ∂_ttX=∂_z[(ρ gz + Mg) ∂_zX ],

where X is the x-direction component of 𝐗. Through separation of variables X=p(t)q(z)
and assuming that the hanging mass is much larger than the mass of the cord, one finds

    ∂_ttp=-ω^2_np.

The boundary conditions imposed on the cord determine exactly the value of ω_n. If
we assume now that the hanging mass experiences a time-dependent form, and that the gravitational
constant is thus given as g(t)=g[1+a(t)], we find through the same procedure as above

    ∂_ttp=-[1+a(t)]ω^2_np,

where a=F(t)/(Mg), and F is the unknown vertical force exerted on the hanging mass M. An
important assumption was made above, namely that a separation of time and space variables is possible, which is no
longer the case for large tip-mass displacements. Additionally, Eq. (<ref>) is a significant
simplification considering the nontrivial BCs at the lower end <cit.>.
Ideally, one would have to also include the remote force exerted by the plate on the magnet into the bottom
boundary condition, notably in the acceleration of the center of mass.





§ PIVOTING-MASS LENGTH EFFECTS

We now measure experimentally the resonant frequencies
f_0 for different lengths ℓ as shown in Fig. <ref>. A strong dependence of
f_0 on ℓ is found, and f_0 is significantly larger (by one order of magnitude) than the
corresponding usual rigid pendulum mode (2 Hz) or the no-tip-mass flexible string mode (2.4
Hz). The numerical computation of f_0 using Eq. (<ref>) and the BCs found in
Ref. <cit.> is shown as well, which roughly agrees with the data with no fitting
parameter. We thus find that the tip-mass flexible string is reasonably well modeled by Eq. (<ref>) with the BCs 
in Ref. <cit.>.  As
the string is not able experimentally to freely rotate around its lower attachment point (see inset
of Fig. <ref>), a rigid motion occurs on the string δℓ≃2 mm away from its
lower end, and we have thus introduced an effective length of the axis, ℓ=ℓ+δℓ,
in this model.



§ HILL'S EQUATION

Let us now take into account the remote magnetic forcing as the vertical oscillations of the
conducting plate induce a time-dependent force on the magnet. First, we measure, with no forcing and
using a Koshava5 Gaussmeter, the field amplitude B_x(z) generated by the magnet along the negative
z axis (i.e., below the magnet and in the direction of the magnetic dipole moment
𝐦). Far from the magnet, B_x(z) is well described (see inset of
Fig. <ref>) by the classical magnetic dipole law
B_x(z)=μ_0 m/(4π z^3) <cit.>, with μ_0=4π10^-7 Tm/A the vacuum
magnetic permeability. By moving closer to the magnet, the experimental magnetic strength clearly
departs from this prediction and is roughly fitted by B_x(z)=μ_0q/(4π z^2) where q is an
effective monopole moment inferred from the fit as q=33± 4 Am. Note that the components B_y
and B_z are negligible due to the choice of 𝐦 along the x direction, since they are measured to be ten times
  smaller than B_x. Additionally, more sophisticated expressions for the field of a permanent magnet could be
  used <cit.>, but for the sake of simplicity we rely on this empirical model,
  also known as the dumbbell model <cit.>. The electromagnetic interaction between the
magnetic field B_x and the vertical motion of the nearby oscillating conductor causes a vertical
force on the magnet computed as F_z(d) = (μ_0^2q^2σ u_z)/(16π d) (see <ref>)
with d(t)=d_0+Asin(2π ft) the relative distance between the magnet and the plate,
u_z=dd/dt their relative vertical velocity, and σ the plate
conductivity. By adding the driving force F_z into Eq. (<ref>) of the hanging string,
including the nontrivial BCs, and assuming small horizontal displacements, we find that the
amplitude p_n of each mode n, oscillating at ω_n, is governed by an equation of the Hill
type (see <ref>)

    ∂_ttp_n=-ω^2_n[1+c(A/d_0)ωcos(ω t)/1+(A/d_0)sin(ω t)]p-κ∂_tp_n

with c=(μ^2_0q^2σ)/(16π Mg)=0.026 s, and κ a dissipation coefficient. It is
well-known that the Hill equation can lead to resonant tongues of parametric instability including the
special case of the Mathieu equation <cit.>.




By applying a Fourier expansion of the periodic function within the brackets in
Eq. (<ref>) up to the second order, we can analytically find the expressions for the
resonant tongues in the (f/f_0, A/d_0) parameter space (see <ref>). The
tongues originate at 2ω_n/m (m=1, 2, ... is the tongue number) leading thus to a harmonic
or subharmonic response. We reproduce here only the expression for the subharmonic tongue at 2f_0, i.e.,

    (2ω^2_0/ω^2-1/2)^2+κ^2/ω^2-(c e_1ω_n^2/ω)^2=0.

where e_1 corresponds to the first Fourier coefficient in the series expansion, which depends
nontrivially on the forcing amplitude (the exact expressions being cumbersome, they have been collected in <ref> and <ref>). The subharmonic instability curve of Eq. (<ref>) is
plotted in Fig. <ref> (red solid line) along with the harmonic prediction at f_0 (blue
one) from Eq. (<ref>). Our model is found to be in good agreement with the experimental instability curves with no
fitting parameter once the tongue minimum is known. As the latter is related to
dissipation <cit.>, this leads to an estimation of the effective dissipation κ=0.18
s^-1 satisfying the timescale separation 1/f ≪ 1/κ. Slight deviations of the model for higher frequencies (compare red solid line and bullets) could be due to the mode truncation to obtain the analytic expression. In addition, the theoretical tongue shape near f_0 (blue solid line) exhibits a sharp, almost vertical, rise above f_0, explaining why we are unable to experimentally observe the
right part of the harmonic instability curve.























§ TRANSITION BEHAVIOR

We finally focus on the magnet dynamics when crossing the subharmonics instability threshold. For this, we increase the plate amplitude A at a fixed frequency f, and, using the laser vibrometer, we measure the horizontal velocity v(t) of the magnet-string
  motion, as shown in top Fig. <ref>. For low A, small-scale oscillations are reported
  (see blue curve). For high enough A, still below the subharmonic instability threshold A_c(f) in
  Fig. <ref>, we observe the appearance of higher harmonics as well as an asymmetry of
  the motion on every second period of the forcing frequency. This asymmetry is likely because the magnet velocity is no longer negligible. Once we cross the threshold of instability, the motion becomes more regular and with an
  amplitude that is ten times larger, as seen in bottom Fig. <ref>, with period doubling
  clearly visible.  Interestingly, the magnet spends a long time at its extreme points, i.e., for v=0, with
  a rapid pass through its equilibrium point. Thus, increasing the amplitude means that higher harmonics grow
  more and more until the system transitions into the period-doubling state. This
  behavior is similar to that of the Duffing equation and indicates the presence of nonlinearities
  in the string.

 


§ CONCLUSION

We have reported, both experimentally and theoretically, the dynamics of an articulated oscillating
  system consisting of a flexible string with a freely pivoting mass at its end. The system resonant
frequency is controlled by the length of the pivoting mass and differs significantly (by one order
of magnitude) from that of the single flexible string. This articulated system in
  particular  exhibits bending oscillations, similar to a swing becoming slack. To do so, we used a
  remote and noninvasive parametric forcing technique. A magnet suspended by a flexible string
above a vertically oscillating conductive plate is subjected to a remote force (due to induced
  currents in response to the conductor motion). Parametric resonances of the tip-mass-string
system are then highlighted such as harmonic and period-doubling instabilities. This parametric system is theoretically modeled using a Hill equation and the obtained analytic expressions for the resonant tongues describe well the observed instabilities. Beyond their
thresholds, more complex dynamics can be observed once the magnet displacement or velocity becomes
comparable to the oscillating plate
one. A potential improvement to the current study would be to consider different magnet strengths. 



























§ ACKNOWLEDGMENTS

We thank V. Leroy for discussions and A. Di Palma, Y. Le Goas, and O.-S. Souramasing for technical help. This work was supported by the Simons Foundation MPS No. 651463–Wave Turbulence (USA) and the French National Research Agency (ANR Sogood Project No. ANR-21-CE30-0061-04 and ANR Lascaturb Project No. ANR-23-CE30-0043-02).









§ MOVIE


 Video of the string, swinging back and forth at half the driving frequency, and showing a period-doubling instability. f=30 Hz.
 https://ars.els-cdn.com/content/image/1-s2.0-S0167278924001155-mmc1.mp4See the video




§ FORCE ON THE MAGNET

In our model, we consider that the magnet is the source of a simplified magnetic field close to the
conducting plate in the form of

    B⃗=μ_0/4πq/r^2x̂,

which we find to be a reasonable approximation empirically. Following Ref. <cit.>, the force on
the source of this field can be obtained through

    F⃗ = ∫_Σσ(u⃗×B⃗)×B⃗V,

where Σ is the conductor volume, considered to be located at a distance d from the source
and filling the half-infinite volume (we find numerically that effects of finite size can be
neglected due to the 1/r^2 drop-off of field strength).

Inserting Eq. (<ref>) into Eq. (<ref>) and considering u⃗=uẑ, Eq. (<ref>) reads

    F⃗ = (μ_0q/4π^2)σ u ∫_Σ(ẑ×ŷ)×ŷ/r^4 V.

By turning to cylindrical coordinates, we have

    F⃗ = (μ_0q/4π)^2σ uẑ∫_d^∞∫_0^∞∫^2π_0 ρ/ρ^2+z^2 ϕρz,

finally yielding

    F⃗ = (μ_0q/4)^2σ u/π dẑ.




§ HILL EQUATION AND STABILITY

We can now insert Eq. (<ref>) into Eq. (<ref>) to find the
governing equation of a single mode as

    ∂_ttp=-ω^2_n[1+(μ_0q/4)^2σ/π Mg Aωcos(ω t)/d_0+Asin(ω t)]p,

where we have assumed that the conducting plate position is given by z(t)=Asin(ω t) and its
motion is larger than the magnet one, much like the linearized equations of motion found in Ref. <cit.>.

We introduce now τ=ω t and perform a change of variables

    ∂_ττp=-[α+γ f(τ)]p,

where

    α   =ω^2_n/ω^2,    γ = (μ_0q/4)^2σω_n^2/πω Mg
    
      f(τ)   =βcos(τ)/1+βsin(τ),   β=A/d_0.

In order to analytically compute the approximative instability regions of Eq. (<ref>), we expand the
function f(τ) into a Fourier series up to the second order. This yields

    ∂_ττp=-[α+γ(e_1cosτ+e_2sin 2τ)]p-κ∂_τ p,

where a damping term κ is now included. Both e_1 and e_2 are functions of β
and are given in Eq. (<ref>). We consider p=∑_-∞^∞ a_n e^inτ/2, giving us a system of
equations for a_n. We truncate the system to two dimensions and impose that the corresponding
determinant is zero, i.e.,

    α-1/4+i/2κ   1/2γ e_1 
    1/2γ e_1    α-1/4-i/2κ

giving

    (α-1/4)^2+1/4κ^2-1/4(γ e_1)^2=0.

We repeat the procedure for the harmonic solutions with p=∑_-∞^∞ a_n e^inτ, except that a larger
dimension is needed now

    2(α-1+iκ)    γ e_1     -iγ e_2 
    γ e_1     2α   γ e_1 
    
    iγ e_2    γ e_1     2(α-1-iκ)

which leads to

    α[4(α-1)^2+4κ^2-γ^2e_2^2]-2(α-1)γ^2e_1^2=0.




§ FOURIER COEFFICIENTS

The function f(τ) is expanded as

    f(τ)≈ e_1cos(τ)+e_2sin(2τ)

with

    e_2 =   2/β^2(β^2+2 √(1-β^2)-2) 
    
      e_1 =   r/2πβ√(1-β)
    
      r =    4 π(2 √(β+1)β+√(1-β) -2 √(β+1))  
        -8 (β-1) √(β+1)tan ^-1(√(1-β/β+1))   
        -8 (β-1) √(β+1)tan^-1(√(β+1/1-β))




§ DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.



§ DATA AVAILABILITY

Data will be made available upon request




100

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