Ponytail Motion
Author: Joseph B. Keller
Source: SIAM J. Appl. Math
Vol. 70. No. 7, pp. 2667-2672
About Author:
Joseph Bishop Keller
Born
Residence
Nationality
Fields
Institutions
Alma mater
Known for
July 31, 1923 (age 89)
Paterson, New Jersey
U.S.
American
Mathematician
New York University
Stanford University
New York University
Geometrical Theory of Diffraction
Einstein–Brillouin–Keller method
Notable awards
 National Medal of Science (USA) in Mathematical, Statistical, and Computational
Sciences (1988)
 Wolf Prize (1997)
 Nemmers Prize in Mathematics(1996)
This presentation will:
Demonstrate the motion of a ponytail
 Analyze the stability/instability of this motion
Key words and Quick definitions:

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





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


Frequency:
The number of occurrence of a repeating event(or cycles) per unit time.
Angular Frequency:
Angular frequency(or angular speed) is the magnitude of angular velocity
Amplitude:
The measure of change in a periodic variable over a single period (or peak deviation from zero).
Oscillation:
A repetitive variation typically in time of some measure about a central value or between two or more states.
Resonance:
The tendency of a system to oscillate with greater amplitude at some frequencies than at others.
Parametric Resonance:
The phenomenon of resonance that deals with the instability conditions.
Linearization:
Finding a linear approximation to a function at a given point.
Exponential Growth:
When the growth rate of the value of a mathematical function is proportional to function’s current value.
Excitation:
An elevation in energy level above an arbitrary baseline energy state.
Parametric Excitation:
The method of exciting and maintaining oscillations in a dynamic system in which excitation results from a periodic
variation in energy storage element in a system e.g. excitation of swing due to properly time bending of the knees.
Equilibrium Point:
The point
is an equilibrium point for the differential equation
if
Perturbation:
A small change/disturbance in the physical state (or initial/existing condition) of a system.
Lateral Perturbation:
A change which occurs in the state side by side as the physical state changes with time.
How to establish the equations of motion…??
Equations of motion are formulated as a system of second-order
ODE’s that may be converted to a system of first-order equations
whose dependent variables are positions and velocities of the
object.
Generic form for such systems:
  f (t , x),t  0, x(0)  x0 --------(a)
x
where:
• x0 is a specified initial condition for the system
• the component of x are the positions and velocities of the object
• f (t , x) includes the external forces and torques of the system
Example:
The equation of motion of pendulum    g sin   b 
l
lm

x


,
x


2
Replace ( 1
)
g
c
x2   sin x1 
x2
l
lm
For linearization replace (sin x  x )
How we discuss Stability and Instability?
Stability:
A solution  (t ) to the system (a) is said to be stable if every
solution  (t ) of the system close to  (t ) at initial time t  0 remains
close to all future time.
In mathematical terms, for each choice of
  0 such that  (t )   (t )   whenever
  0 there is a
 (0)   (0)  
Instability:
If for at least one solution  (t ) does not remain close, then  (t ) is
said to be unstable
Eigenvalues can also be helpful:
Let A  Df (c) be the matrix of first-order partial derivative of f (x)
(Jacobian Matrix)evaluated at c, then:
 Every solution(or equilibrium) is stable if all eigenvalues of A
has negative real parts.
 Every solution(or equilibrium) is unstable if at least one
eigenvalue of A has positive real part.
Hill’s equation (G. W. Hill 1886) can also be helpful:
The hill’s equation is a second order linear ODE,
d2y
 f (t ) y  0
d t2
where f (t )is a periodic function.
If Hill’s equation has the solution that grows exponentially with
time then motion will be considered as unstable.
Few reminders:
The ponytail of the
running jogger
sways from side to
side
The swaying
(lateral motion) is
an example of
parametric
excitation
Jogger’s head
generally
moves up and
down
The vertical motion
of hanging ponytail
is unstable to lateral
perturbations
Suggested ways to study this motion:
• Either consider the ponytail as
a rigid pendulum
• Or consider the ponytail as a
flexible string
• Or consider the ponytail as an
inextensible rod with small
bending stiffness
1st Case: Ponytail as a rigid rod
As runner moves along +z-axis, her head moves
up and down along y-axis.
One end of ponytail is attached to jogger’s head at
the position x  o, y  a(t ), z  Ut
Consider,
L = Length of ponytail (a uniform rigid rod)
U = runner’s speed along z-axis
= Position of the ponytail in the plane z=Ut
a(t) = periodic vertical displacement along y-axis  A cos t
A = amplitude of oscillation
Then it is a simple pendulum having one end point fixed
with vertical acceleration
added to the acceleration
due to gravity.
2
tt  ( g  att ) sin  0 -----(1)
L
With the vertical acceleration att of the end point added to the acceleration of
gravity g, eq.(1) has two solutions in the interval 0 < < 2:
1. 0  0 , means pendulum hanging straight down
2. 0   , means pendulum balanced pointing upward
The Stability/instability of either solution determined by the equation for

perturbation  (t ) obtained by linearizing about 0 :
2
---------(2)
tt  ( g  att )  0
L
Which shows that system oscillates between limits [0 ,0 ] :

,
When att  0 :
the solution for  is sinusoidal for 0  0
the solution for  is exponentially growing or decaying for 0  
means the hanging pendulum is stable and the balanced pendulum is unstable
 when att  0 (but a periodic function of t, eq.(1) is called Hill’s equation):
Recall:
Equation of motion for a simple pendulum:
Hill’s equation:
And
Where
tt 
2
( g  att ) sin  0
L
d2y
 f (t ) y  0
d t2
If Hill’s equation has the solution in the interval of
that grows
exponentially with time then motion will be considered as unstable.
is the result of equation of motion in dimensionless parameter.
Mathematically:
For any periodic function a(t ) with frequency  there are infinite many intervals of
throughout which Hill’s equation has solutions that grows exponentially with t.
In Ponytail situation:
When the solution lies in one of these intervals, the hanging pendulum becomes
unstable or we observe the swaying of ponytail.
Few interesting calculations:
For a ponytail of length L=25cm has natural frequency
(2  980/ 25)1/ 2  8.85radians/ sec  1.41cycles/ sec
must have the frequency of motion of jogger’s head twice the natural frequency means
  17.71radians/ sec  2.82cycles/ sec
A cycle correspond to a step with one leg means
2.82cycles/ sec  169cycles/ min  169steps / min
Summary of case 1
A ponytail of length 25 cm can be expected to sway at a typical running cadence which is
160 steps/min according to website RunGearRun.com
2nd Case: Ponytail as a flexible string
Let the ponytail hanging in the plane z  Ut having:
L = Length of ponytail as inextensible flexible string
 = constant density of string
T = tension in the string
g = (0,-g) = acceleration
Let x( s, t )  ( x( s, t ), y( s, t ))be the position at time t of the point at arc-length
distance s from the top of string then:
 x  (T x )   g
 it satisfies equation of motion:
0<s<L --------(3)
The condition that s is arc-length requires: x  1
0<s<L --------(4)
x(0, t )  (0, a(t ))
 Position in the plane at the end s=0:
--------(5)
T ( L, t )  0
 Tension vanishes at the end s=L:
--------(6)
tt
s s
2
s
One solution of eq.(3)-eq.(6) represents vertically hanging string moving up and down is:
---------(7)
x (s, t )  [0, a(t )  s]
0
and the corresponding tension is:
T 0 (s, t )   ( g  att )(L  s)
---------(8)
Again for checking Stability/Instability we need to see the linearized problem for perturbation
0
0

x
T
in x and by linearizing around the solution and T , Which will become:

 x tt  (T x0s  T 0 x s )s  Ts y  (T 0 x s )s
 Equation of motion:
The condition that s is arc-length requires:
 Position in the plane at the end s=0:
 Tension vanishes at the end s=L:
0<s<L ---------(9)
0<s<L -------(10)
--------(11)
--------(12)
0
xs  x s   ys  0
x (0, t )  0
T ( L, t )  0
Integrate eq.(10) w.r.t s and using the y-component of eq.(11) gives:
When eq.(13) used in y-component of of eq.(9) gives:
or
------(13)
------(14)
Solution for lateral displacement:
 when eq.(8) is used for T 0 , the x-component of eq.(9) becomes x   ( g  a )[(L  s) x ] ---(15)
 ( s, t )  u(t )v( s)
x
 we are interested in solution having product form
---(16)
 substitution of eq.(13) into eq.(14) gives
---(17)
u ( g  a ) u  [(L  s)v ] v  
 from eq.(14) we get two equation :
[(L  s)v ]  v  0
--------(18)
and
--------(19)
u   ( g  a )u  0
with boundary condition v(0)  0.
tt
tt
1 1
tt
s s
tt
tt
tt
1
s s
s s
The only solution for eq.(18) which is regular at s=L is a constant multiple of Bessel’s
v(s)  J 0[21/ 2 ( L  s)1/ 2 ]
function J 0 :
--------(20)
we call this solution for nth mode v (s) and substitute in eq.(16), then the desired result
x(s, t )  u(t , n ) J0[(1  s / L)1/ 2 jn ] --------(21)
will be:
n
The amplitude u (t , n ) in eq.(21) satisfies eq.(19), which is Hill’s equation with
Mathematically:
For any periodic function a(t ) with frequency  there are infinite many intervals of
n g /  2 throughout which Hill’s equation has solutions that grows exponentially with t.
In Our situation:
when solution lies in one of these intervals, the vertical motion of the
flexible string becomes unstable to the lateral perturbations or we
observe the swaying of ponytail.
Few interesting calculations:
For the lowest mode n  1 and j1  2.4 , the mode frequency is j1 / 2( g / L)1/ 2  1.2( g / L)1/ 2
For a ponytail of length L=25 cm when
 is around twice the lowest mode frequency i.e.,
2(1.2)(980/ 25)1/ 2  15.0radians/ sec  2.39cycles/ sec.  143.5steps / min.
Summary of case 2
143.5 steps/min is slightly less than the cadence required for swaying the jogger’s ponytail
having length 25cm but still the ponytail can be expected to sway.
A more realistic model
Ponytail as a inextensible flexible rod
When runner is not moving the ponytail will extend away
from head and hang downward in its characteristic shape.
e.g. Cantilever Beam
When runner is moving and her head is bobbing up and down
and ponytail oscillate in yz-plane, the instability of this motion
would determine when swaying occurs and would determine
the swaying mode shape.
The equation of motion with the addition of bending term  Bxssss
xtt   ( g  att )[(L  s) xs ]s  Bxssss
will become:
Since it is of fourth order, so it needs four boundary condition:
s (0, t )  0
 (0, t )  0, x
x
Two conditions for the ponytail clamped at the top
ss ( L, t )  0 , x
sss ( L, t )  0
x
Two conditions for the ponytail free at the bottom
References:
1] J. J. Stoker, Nonlinear Vibrations, Interscience, New York, 1950.
[2] W. Magnus and S. Winkler, Hill’s Equation, Interscience, New York, 1966.
[3] A. Belmonte, M. J. Shelley, S. T. Eldakar, and C. H. Wiggins, Dynamic patterns and
self-knotting of a driven hanging chain, Phys. Rev. Lett., 87 (2001), pp. 114301–114304.
[4] A. Stephenson, On a new type of dynamical stability, Mem. Proc. Manch. Lit. Phil. Soc., 52
(1908), pp. 1–10.
[5] D. J. Acheson, A pendulum theorem, Proc. Roy. Soc. London Ser. A, 443 (1993), pp. 239–245.
[6] D. J. Acheson and T. Mullin, Upside-down pendulums, Nature, 366 (1993), pp. 215–216.
[7] G. H. Handelman and J. B. Keller, Small vibrations of a slightly stiff pendulum, in Proceedings
of the 4th U.S. National Congress on Applied Mechanics, Amer. Soc. Mech. Eng., New
York, 1963, pp. 195–202.
[8] A. R. Champneys and W. B. Fraser, The “Indian rope trick” for a parametrically excited
flexible rod: Linearized analysis, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456
(2000), pp. 553–570.
Note: Graphics and images used in this presentation are easily available on Google images section.
Note: Quick definitions used in this presentation are taken from mathematics section based websites.
Presented by: Adnan Ahmed
Few additional:
Newtonian Equation of Motion:
Hamiltonian Equation of Motion:
Bessel’s Differential Equation:
Bessel’s functions are canonical solutions y(x) of Bessel’s differential equation
Where Γ is the gamma function, a shifted generalization of the factorial
function to non integer values.
Parametric Excitation:(Journal of Applied Physics, vol 22, num. 1, Jan 1951)
If a parameter of an oscillatory system is varied periodically between certain
limits, the system become excited, i.e., start oscillating with frequency equal
to one-half of that with which the parameter varies. The term parametric
excitation is used to designate this phenomenon.
The stability of upside down pendulums:
Theorem:
Let N pendulums hang down, one from another, under gravity g, each having one degree
of freedom, the uppermost being suspended from a pivot point O. Let ω(max) and ω(min)
denote the largest and the smallest of the natural frequencies of small oscillation about
this equilibrium state.
Now turn the whole system upside-down. The resulting configuration of the pendulums
can be stabilized (according to linear theory, atleast) if we subject the pivot point O to
vertical oscillations of suitable amplitude Є and frequency . When
the stability
criterion is
------ (1)
NOTE: when several pendulums are involved
is typically much greater than
.
The condition
is then necessary for the stability of the inverted state. So eq(1)
then gives the whole stability region in the Є- plane.