Further insights into the damping-induced self-recovery
phenomenon
Tejas Kotwal ∗, Roshail Gerard †and Ravi Banavar

‡

arXiv:1709.05596v4 [] 11 Jan 2018

January 12, 2018

1

Abstract

In a series of papers [1, 2, 3, 4], D. E. Chang, et al., proved and experimentally demonstrated a phenomenon they termed “damping-induced self-recovery”. However, these papers left a few questions
concerning the observed phenomenon unanswered- in particular, the effect of the intervening lubricantfluid and its viscosity on the recovery, the abrupt change in behaviour with the introduction of damping,
a description of the energy dynamics, and the curious occurrence of overshoots and oscillations and
its dependence on the control law. In this paper we attempt to answer these questions through theory. In particular, we derive an expression for the infinite-dimensional fluid-stool-wheel system, that
approximates its dynamics to that of the better understood finite-dimensional case.

2

Introduction

The damping-induced self-recovery phenomenon refers to the fundamental property of underactuated
mechanical systems: if an unactuated cyclic variable is subject to a viscous damping-like force and the
system starts from rest, then the cyclic variable will always recover to its initial state as the actuated
variables are brought to rest. A popular illustration exhibiting self-recovery is when a person sits on
a rotating stool with damping, holding a wheel whose axis is parallel to the stool’s axis. The wheel
can be spun and stopped at will by the person (Refer to [5]). Initially, the system begins from rest;
when the person starts spinning the wheel (say anticlockwise), the stool begins moving in the expected
direction (i.e., clockwise). Then the wheel is brought to a halt; the stool then begins a recovery by
going back as many revolutions in the reverse direction (i.e., anticlockwise) as traversed before. This
phenomenon defies conventional intuition based on well-known conservation laws. Andy Ruina was the
first to report this phenomenon in a talk, where he demonstrated a couple of experiments on video
[6]. Independently, Chang et al., [1, 2] showed that in a mechanical system with an unactuated cyclic
variable and an associated viscous damping force, a new momentum-like quantity is conserved. When
∗ Department

of Mathematics
of Mechanical Engineering
‡ Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai, India
† Department

1

the other actuated variables are brought to rest, the conservation of this momentum leads to asymptotic
recovery of the cyclic variable to its initial position. Boundedness is another associated phenomenon, in
which the unactuated cyclic variable reaches a saturation eventually, when the velocity corresponding to
the actuated variable is kept constant; this occurs due to the presence of damping. In the experiment
explained above, this manifests as the angle of rotation of the stool reaching an upper limit, when the
wheel is spinning at a constant speed.
Chang et al. generalize this theory to an infinite-dimensional system in which the interaction of an
intermediate fluid is considered [3, 4]; they show that the fluid layers also display self-recovery, which
is confirmed via experiments as well [7]. Such a generalization of the model is considered because of
the interaction of the fluid in the bearing with the recovery phenomenon of the stool, in the experiment
explained above. In this work, we make the following points:
• We show that the dynamics of the stool and the wheel in the infinite-dimensional fluid system can
be approximated to that of the finite-dimensional case by finding an effective damping constant
that takes into account the effect of the viscous fluid on the system.
• We analyse the finite-dimensional system from a dynamical systems point of view, and show that
a bifurcation occurs when the damping constant switches from zero to a positive value. We also
derive an expression for the angle at which boundedness occurs.
• In addition to the recovery phenomenon described previously, further complex behaviour is observed
in the experiments reported in [5, 7]. In particular, the unactuated variable not only approaches
its initial state during recovery, but also overshoots and then oscillates about this initial position,
eventually reaching it asymptotically. This oscillation phenomenon has not been looked into in
previous works, and is one of the points that we address as well.
• In Chang et al. [5], in the experiment described, the oscillations are of significant amplitude, and
this would prompt one to assume that some sort of mechanical ‘spring-like’ energy is being stored,
as the stool appears to start moving after the entire system has come to a halt. The question of
energy has been touched upon in this work, and we present energy balance equations for the given
mechanical system.
The paper unfolds as follows. Initially we present mathematical models for the stool-wheel experiment
- the first is a finite-dimensional one, and the second one incorporates the intervening fluid (either in
the bearing or in a tank) using the Navier-Stokes equation for a Newtonian incompressible fluid. This
is followed by a section that presents an intuitive interpretation of recovery highlighting three distinct
types of behaviour. Then follows a theoretical section that presents a technique to reduce the infinitedimensional fluid effect to an effective damping constant and hence model the overall system in finite
dimensions. This part is followed up by a result on boundedness and the occurrence of a bifurcation
in the system dynamics. In the appendix, the derived results are used in conjunction with numerical
experiments to validate the expression for the effective damping constant. We then examine the case of

2

oscillations and overshoots, and present plausible explanations to why these occur, and possible sources
of future investigation.

3

Mathematical models

Finite-dimensional model: We first analyze a simplified, idealized version of the person with a wheel
in hand, sitting on a rotatable stool whose motion is opposed by damping, which for the purpose of
analysis is assumed to be linear viscous damping. This is a specific example of the model that Chang, et
al. studied [1, 2]. We assume two flat disks, one for the wheel and one for the stool-person mass as shown
in Fig. 1. The stool consists of an internal motor, that actuates the wheel, while the motor-rod-stool
setup rotates as one piece (henceforth just called the stool). There is linear viscous damping present in
the rotational motion of the stool, with damping coefficient k. The torque imparted on the wheel by the
motor is denoted by u(t).

Figure 1: A schematic diagram of the wheel-stool model.
The inertia matrix of the system is given by

(mij ) = 

m11
m21


I
= w
m22
Iw

m12



Iw
Iw + Is


,

(1)

where Iw and Is are the moments of inertia of the wheel and stool respectively. The kinetic energy
of the described system is

K.E.(t) =

1
1
Iw (θ̇w + φ̇s )2 + Is φ̇2s ,
2
2

(2)

where θw denotes the angle rotated by the wheel relative to the stool, and φs denotes the angle
rotated by the stool relative to the ground frame. Since there is no external potential in our system, the

3

Lagrangian only comprises of the total kinetic energy. The Euler-Lagrange equations for the system are
given by

Iw φ̈s + Iw θ̈w = u(t)

(3a)

(Iw + Is )φ̈s + Iw θ̈w = −k φ̇s

(3b)

Although this model captures the damping-induced boundedness and recovery phenomena, further
behaviour that cannot be explained with this model are the overshoot and oscillations, which were
observed in experiments [5, 7]. It was suggested that the lubricating fluid inside the bearing of the
rotatable stool could be one of the reasons behind this additional behaviour. As a result, a generalized
model that involves the dynamics of the fluid interacting with the stool-wheel setup is analyzed [3, 4].
Infinite-dimensional fluid model: The model consists of two concentric infinitely long cylinders
with inner radius Ri and outer radius Ro . The outer radius is fixed, while the inner cylinder is free
to rotate (this is the stool in our case). A motor is installed in the inner cylinder (as described in the
previous case, i.e., φs ), to drive the wheel (i.e., θw ) attached to it. The annulus region between the two
cylinders is filled with an incompressible viscous fluid. Due to symmetry in the z direction, we may
regard this as a 2D system in a horizontal plane. The moments of inertia should be considered as per
unit depth, because of the infinite length in the z direction. Due to rotational symmetry, the fluid only
flows coaxially, i.e, there is no fluid flow in the radial direction. Let v(r, t) denote the tangential velocity
of the fluid at radius r and time t, where Ri ≤ r ≤ Ro . The subscripts t and

r

denote partial derivatives

with respect to time and radius respectively. As described before, u(t) is the driving torque on the wheel
imparted by the motor. The corresponding equations are

Iw φ̈s + Iw θ̈w = u(t)

(4a)


(Iw + Is )φ̈s + Iw θ̈w = 2πρνRi Ri vr (Ri , t) − v(Ri , t)

(4b)

vt = ν(vrr +

vr
v
− 2)
r
r

(4c)

where ν is the kinematic viscosity and ρ is the density of the fluid. The right hand side of Eq. (4b)
is the torque on the inner cylinder due to stress exerted by the surrounding fluid [8]. Equation (4c) is
the Navier-Stokes equation for an incompressible viscid fluid in radial coordiantes. The system (4) is
an infinite-dimensional fluid system, compared to the previous finite-dimensional system (3). The initial
and boundary conditions are given by

φs (0) = φ̇s (0) = θw (0) = θ̇w (0) = 0

(5a)

v(r, 0) = 0

(5b)

v(Ri , t) = Ri φ̇s (t)

(5c)

v(Ro , t) = 0

(5d)

4

where Eqs. (5c) and (5d) are the no-slip boundary conditions for the fluid. The given initial conditions
imply that the entire system begins from rest. In order to control the wheel independently of the effects of
inertia of the stool and damping of the fluid, we employ a method known as partial feedback linearization.
This is done by partially cancelling the nonlinearities in the dynamics, i.e. the wheel equations are
linearized by introducing a new input τ (t) and redefining the input u(t) as

u(t) =

Iw
Iw + Is


Is τ (t) + 2πρνRi Ri vr (Ri , t) − v(Ri , t)



(6)

where τ (t) is precisely equal to the acceleration of the wheel, i.e. θ̈w = τ (t). We assume a PD control
law for the new control variable given as



d
d
d
τ (t) = θ̈w
(t) + c1 θ̇w
(t) − θ̇w + c0 θw
(t) − θw

(7)

d
where θw
(t) is the desired trajectory of the wheel, and c0 and c1 are the proportional and differential

gains respectively. For the standard demonstration of recovery and/or boundedness, we require the wheel
to be driven from rest to constant velocity, and then abruptly braked to a stop. Such a desired trajectory


d
(t) = (1/2) θ̇steady |t| − |t − tstop | + tstop where θ̇steady is
of the wheel is given by a ramp function, θw
the desired constant velocity of the wheel, and tstop is the time at which the wheel is instantaneously
brought to rest (Refer to Fig. 2). By appropriately tuning the values of c0 and c1 , we can mimic the
desired trajectory. This tuning is important in accounting for the presence of oscillations, or lack thereof,

Desired trajectory of the wheel 3wd (rad)

as we shall see later.

slope = 3_steady

0

0

tstop

Time t (s)

d
Figure 2: An illustration of the desired trajectory θw
(t) in the control law of the wheel (7)
.

4

An intuitive interpretation of recovery

We present a qualitative interpretation of the boundedness and recovery phenomenon using the momentum equation of the finite dimensional stool-wheel case. A brief explanation of the recovery phenomenon
was also presented by Andy Ruina [6] which we summarize: Change in angular momentum is equal to

5

the net external torque, but since it is only due to linear viscous damping, we have L̇ = −cϕ̇ (where L is
the angular momentum, c is the damping constant and ϕ is the angle). If the system’s initial and final
state is the rest state, then ∆L = 0 =⇒ ∆ϕ = 0. Thus the net change in angle has to be zero, implying
that recovery must occur. We delve a little deeper into the phenomenon and attempt to provide an
intuitive interpretation for the reversal of direction of the stool.
Consider the case where the wheel follows the desired trajectory perfectly (i.e., the ramp function).
This means that at t = 0, an impulsive acceleration gets the wheel spinning instantaneously (to a constant
speed), and at t = tstop , an impulsive deceleration brings the wheel to a halt instantaneously (refer to
Fig. 3). Consider the damping-induced momentum equation, as derived by Chang, et al. [1, 2]

Acceleration of wheel 3Bw (rad=s2 )

Iw (θ̇w + φ̇s ) + Is (φ̇s ) + kφs = 0

(8)

2000
1000
0
-1000
-2000
0

2

4

6

8

10

Time t (s)

Figure 3: Acceleration vs time for the wheel. The first peak corresponds to the initial impulse to get the
wheel spinning to a constant velocity; the second negative impulse is to brake the wheel to a halt.
At t = 0, there is a jump in θ̇w , while φs remains constant (i.e., φs = 0) (refer to Figs. 8 and 9). The
momentum equation (8) then simplifies to

Iw (θ̇w (0) + φ̇s (0)) + Is (φ̇s (0)) = 0

(9)

which is the usual momentum conservation. Thus, at t = 0, the velocity of the stool is governed
by the conservation of standard angular momentum, and the velocity of the wheel. Let this velocity be
φ̇s (0) = φ̇fs .
At t = tstop , there is an instantaneous jump down to zero for θ̇w , while φs remains constant. This
time, the momentum equation (8) simplifies to

Iw (θ̇w (tstop ) + φ̇s (tstop )) + Is (φ̇s (tstop )) = C

(10)

where C is a constant (C = −kφs (tstop )). Once again, this equation may be interpreted as the
standard angular momentum conservation. We examine three distinct cases:
• Case 1: (No damping) The speed of the stool at t = tstop − ε is the same as what it was at t = 0,
6

(i.e. φ̇fs ), implying that at t = tstop , the stool comes to rest instantaneously (i.e., φ̇s (tstop ) = 0).
Recall, that this is the standard case, where the usual momentum conservation law holds. Refer
to Figs. 7 and 4.

Position of stool ?s (rad)

0
-0.2
-0.4
-0.6
-0.8
-1

0

2

4

6

8

10

Time t (s)

Figure 4: Case 1. k = 0, tstop = 5, vw = 2. This case corresponds to the usual momentum conservation.

• Case 2: (Damping present but damping induced boundedness not yet reached) During 0 < t <
tstop , the damping force keeps decreasing the speed of the stool. However, the momentum equation
(10) is still the same at t = tstop . Thus, the change in momentum of the stool would be the same (as
in Case 1). But since |φ̇s (tstop − ε)| < |φ̇fs |, the final momentum of the wheel ends up overshooting
zero at t = tstop , resulting in the change in direction of the stool. Refer to Figs. 7 and 5.
• Case 3: (Damping present and damping-induced boundedness has been attained) This case can
be viewed as a special instance of Case 2, where for 0 < t < tstop , the speed of the stool decreases
all the way to zero. The rest of the explanation follows as in Case 2. Refer to Figs. 7 and 6.

7

Position of stool ?s (rad)

0
-0.02
-0.04
-0.06
-0.08
-0.1

0

2

4

6

8

10

Time t (s)

Figure 5: Case 2. k = 1, tstop = 1, vw = 2. In this case, damping-induced boundedness is not yet
reached, but the recovery phenomenon can be seen.

Position of stool ?s (rad)

0

-0.05

-0.1

-0.15

0

2

4

6

8

10

Time t (s)

Figure 6: Case 3. k = 1, tstop = 5, vw = 2. In this case, both—damping-induced boundedness and
recovery phenomenon can be seen.

8

Change in velocity of the stool
before and after braking

0

Case 1

Case 2

Case 3

Figure 7: Comparison of the three cases before and after braking. Blue dots: velocity of stool just
before braking; red dots: velocity of stool immediately after braking. For cases 2, 3, the (magnitude of)
velocity of the stool before braking decreases due to damping; hence momentum transferred by the wheel
overshoots the stool velocity to something positive. This explains why the wheel changes direction.

9

5

Effective damping constant

In this section, we demonstrate that the dynamics of the infinite-dimensional fluid system can be approximated by that of a finite-dimensional one.
d
Claim 5.1. Let the desired trajectory of the wheel be the ramp function, given by θw
(t) = (1/2) θ̇steady |t|−


|t − tstop | + tstop . Then the solution of Eq. (4c) satisfies

lim v(r, t) = lim Ri φ̇s (t)

t→∞

t→∞

Ro − r
.
Ro − Ri

Proof: We consider an analytical solution to the PDE (4c), obtained by the method of separation
of variables. Since one of the boundary conditions is non-homogeneous (and in fact time dependent), we
perform a change of variables in order to make it homogeneous [9]. The change of variables is given as

v(r, t) = w(r, t) + Ri φ̇s (t)

Ro − r
Ro − Ri

(11)

where w(r, t) is transformed the variable. The correction term is taken as a linear interpolation
between the two boundary conditions for simplicity. The transformed PDE is given as

wt − ν(wrr +

w
wr
− 2 ) = F (r, t)
r
r

(12)

where F (r, t) is the driving force of the PDE resulting from the change of variables given by

F (r, t) = −Ri φ̈s (t)

Ro − r
Ro
− νRi φ̇s (t) 2
Ro − Ri
r (Ro − Ri )

(13)

The transformed initial and boundary conditions are given by

w(r, 0) = 0

(14a)

w(Ri , t) = 0

(14b)

w(Ro , t) = 0

(14c)

We assume the solution w(r, t) = W (r)T (t), and obtain the homogeneous solution of the spatial ODE
by analyzing the corresponding eigenvalue problem. The eigen solution obtained is given by
r
W (r) = aJ1


r 
λ
λ
r + bY1
r
ν
ν

(15)

where J1 and Y1 are the Bessel’s functions of the first and second kind respectively, λ denotes the
eigen value, and a and b are arbitrary constants determined by the boundary conditions. Upon plugging
in the boundary conditions, and solving for λ, we find that the eigen values are positive due to the fact
that roots of Bessel’s functions are real [10]. Thus we have the following solution for v(r, t)

10

v(r, t) =

∞
X

r
J1

n=1

|


r
!
λn
λn
Ro − r
r + kn Y1
r
Tn (t) +Ri φ̇s (t)
ν
ν
Ro − Ri
{z
}

(16)

transient term

where kn and λn are constants obtained by solving the boundary conditions in the eigen value problem,
and Tn (t) is the time component of the solution, which is obtained by solving the corresponding initial
value problem. The series given in the right hand side of Eq. (16) is considered as a transient term
for the PDE (4c), and we show that it is insignificant as t → ∞. Consider the time component of the
solution (16)
Z

t

e

Tn (t) =
0

R Ro
−λn (t−τ ) Ri

F (r, τ )Wn (r)dr

R Ro
Ri

Wn (r)2 dr

dτ = e

−λn t

Z

t

eλn τ (mφ̈s (τ ) + lφ̇s (τ ))dτ

0

where Wn (r) is the nth term of the Fourier series, and m and l are constants obtained after evaluating
the integrals in the spatial variable. For our case, plug in θ̇w (t) equal to the desired ramp trajectory


d
(θw
(t) = (1/2) θ̇steady |t| − |t − tstop | + tstop ) in the momentum equation (8). Upon solving for φ̇s (t),
we see that it is exponentially decaying, and therefore Tn (t) → 0 as t → ∞ for all n.1 Thus, we have the
result.
.
In the limit where the transient term is insignificant, the solution (16) is plugged in Eq. (4b) and
simplified as follows

(Iw + Is )φ̈s + Iw θ̈w = −

2πρνRo Ri2
φ̇s
Ro − Ri

(17)

We would like to emphasize here that we have found an effective damping constant kef f for the
infinite-dimensional fluid system (4) that approximates its dynamics to that of a finite dimension case
(3). The effective damping constant is given by

kef f =

2πρνRo Ri2
Ro − Ri

(18)

We revisit this effective damping constant and its validity in the appendix, after we introduce a couple
of expressions from the boundedness and bifurcation section.

6

Boundedness and bifurcation

We now revisit the finite dimensional case (3), and rewrite the corresponding momentum equation as a
first order system. The damping-induced momentum equation (8) is given by

(Iw + Is )φ̇s + Iw θ̇w (t) + kφs = 0

(19)

1 In fact, the result holds for any wheel trajectory that grows polynomially for finite time before eventually coming to
rest.

11

where θ̇w (t) is written as a function of time, since it is an external driving force for the stool dynamics.
(Using partial feedback linearization, the wheel can be controlled independently of the stool).
Claim 6.1. Suppose k > 0. Then for a fixed speed of the wheel, say θ̇w (t) = vw = constant, we have the
following equality
lim φs (t) = φ∗s =

t→∞

−Iw vw
k

Proof: Rewritting Eq. (8) as a first order system in φs

φ̇s =

−k
−Iw
φs +
θ̇w (t) = −αφs − β θ̇w (t) = f (φs , t)
Iw + Is
Iw + Is

(20)

where α, β are constants. Since Eq. (20) is a nonautonomous system, we rewrite it in a higher
dimensional autonomous form as

φ̇s = −αφs − β θ̇w (t)

(21a)

ṫ = 1

(21b)

In the differential equation for the stool, for a fixed speed of the wheel, say θ̇w (t) = θ̇steady = vw =
constant, the solution is
Z
φs (t) = −β

t

[e−α(t−s) ]vw ds =

0

−βvw
[1 − e−αt ]
α

(22)

and as t → ∞, we have
φ∗s = lim φs (t) =
t→∞

−βvw
−Iw vw
=
α
k

.
The above equality corresponds to the damping-induced boundedness phenomenon, and is especially
relevant in Figs. 8 and 9. Now we get to the dynamical systems aspect of our discussion.
For the purpose of this discussion we present a few definitions from standard texts on dynamical
systems. A fixed point for a system of differential equations Ẋ = F (X) is defined as the set of points
that are solutions to the equation F (X) = 0. A fixed point for system (21) does not exist since ṫ 6= 0 for
any state (φs , t). We will consider (φ∗s , t∗ ) to be a fixed point if f (φ∗s , t∗ ) = 0. A bifurcation is defined as
a change in the qualitative structure of the flow as the parameters are varied. In particular, fixed points
can be created or destroyed, or their stability can change [11]. In this particular case, the drastic change
in behaviour occurs at k = 0.
d
Claim 6.2. Let the desired trajectory of the wheel be the ramp function, i.e., θw
(t) = (1/2) θ̇steady |t| −


|t − tstop | + tstop . Then

• If k = 0, the stool dynamics is neutrally stable
• If k > 0, the stool dynamics is asymptotically stable.
Thus, a bifurcation occurs when k is varied from 0 to a positive number.
12

Velocity of stool ?_ s (rad=s)

0.1

0
-0.1
-0.2
-0.8

-0.6

-0.4

-0.2

0

Position of stool ?s (rad)

Figure 8: Phase portrait of stool dynamics with varying values of damping coefficient k, and steady
state wheel velocity θ̇steady = 2. Green: k = 1; red: k = 0.1; blue: k = 0. Green: damping-induced
boundedness attained; red: damping induced boundedness is not yet reached; blue: neutral stability, no
recovery phenomenon. The final equilibrium point for the k = 0 case and k > 0 are drastically different.
Proof: k = 0: From eqn. (21),
k = 0 ⇒ φ̇s = −β θ̇w (t)
This implies that the stool dynamics has a fixed point only when the wheel is brought to a rest; the stool
is neutrally stable at any constant value φ∗s .
k > 0: The fixed point φ∗s for the stool dynamics is governed by
φ∗s =

−Iw
θ̇w (t∗ )
k

(23)

For a fixed point to occur, there are two possibilities:
The wheel is at a constant speed vw ⇒ φ∗s = −Iw vw /k

or

The wheel is at rest ⇒ φ∗s = 0

Clearly, if the wheel is braked (zero speed), the stool must go back to its initial state (φ∗s = 0) governed
by
φ̇s = −αφs

(24)

This is the damping-induced self recovery equilibrium state. As seen from the above analysis, the system
undergoes a change in its dynamical system behaviour, when k is switched from 0 to a positive number.
Eq. (23), rewritten as
φ∗s k = −Iw θ̇w (t∗ )

(25)

is the equation of a hyperbola in the variables φ∗s and k, and as k → 0, φ∗s → ∞. For k > 0 (no matter
how small it is), the recovery phenomenon occurs (implying that the stool must return to its initial
position regardless how many rotations it has completed). But for k = 0, the stool is neutrally stable,
implying a change in the stability behaviour. (Note that as φs is 2π−periodic, one should consider

13

Velocity of stool ?_ s (rad=s)

0.3
0.2
0.1
0
-0.1
-0.2
-0.15

-0.1

-0.05

0

Position of stool ?s (rad)

Figure 9: Phase portrait of stool dynamics with varying values of steady state wheel velocity θ̇steady .
Innermost to outermost loop: θ̇steady = 1, 2, 3 (k = 1 for all cases).
φs mod 2π to obtain the actual position of the stool. However, not doing so, gives us the number of
rotations the stool completes. For e.g., if φ∗s = 8π, then the damping-induced bound is attained after
four rotations)
.

7

Oscillations and energy description

Oscillations of the unactuated variable have been reported in a few experiments on the damping induced
self-recovery phenomenon [4, 5, 6, 7]. In our opinion, the cause of these oscillations has not been
adequately identified in previous works. We believe that a seemingly trivial, but important source of
these oscillations, is the nature of the control law used on the wheel. It is observed that if oscillations
are induced in the wheel via the control law, then the stool also mimics these oscillations, with a slight
time lag due to the damping. This type of oscillation can be produced in both, the finite-dimensional
as well as the infinite-dimensional system. Refer to Figs. 10 and 11 for an example via simulations. We
would like to emphasize that oscillations of this kind are a direct consequence of the actuation of the
wheel. It is likely that the oscillations in [4, 7] are of this kind.
However, the oscillations in Ref. [5] occur even though the wheel is stationary with respect to the
stool; in fact, at the extreme positions of the oscillation, it appears as though the entire system (stool
and wheel) begins moving again from a complete state of rest. That is, it appears that there is some
kind of mechanical ‘spring-like’ energy being stored at these extreme positions. These oscillations are
clearly not of the type mentioned above, as they are not a direct result of the actuation of the wheel.
Additionally, the observed oscillations in the stool are of high amplitude whereas those in the wheel, if
any, appear to be of small amplitude. This is contradictory to what is observed in the previous type of
oscillations (seen in Figs. 10 and 11) as the inertia of the stool-wheel system is greater than the inerita
of the wheel only. A simple order of magnitude analysis shows that a more complex model is probably

14

Position of wheel 3w (rad)

500
400
300
200
100
0

0

5

10

15

20

Time t (s)

Figure 10: Trajectory of the wheel (solid blue line) with control parameters c0 = 1 and c1 = 3. The red
dashed line denotes the desired trajectory, and the overshoot of the wheel may be seen at t = 2s.

Position of stool ?s (rad)

0.02
0
-0.02
-0.04
-0.06
-0.08

0

5

10

15

20

Time t (s)

Figure 11: Trajectory of the stool (solid blue line) under the influence of the wheel trajectory shown in
Fig. 10. The black dashed line denotes the zero position of the stool, and the overshoot of the stool may
be seen after it recovers (Note the slight delay when compared to the overshoot of the wheel; this is due
to the time constant of the system).
required to account for this phenomenon. In order to demonstrate this, we first consider the energy
dynamics of the system.
Energy description for the finite-dimensional system: In order to derive the energy equations
of the finite dimensional system, (3b) is multiplied by φ̇s , and (3a) with θ̇w , to yield

(Iw + Is )φ̈s φ̇s + Iw θ̈w φ̇s = −k φ̇s φ̇s

(26a)

Iw φ̈s θ̇w + Iw θ̈w θ̇w = uθ̇w

(26b)

Equations (26a) and (26b) are integrated, and then summed up, resulting in the following energy
balance (in the ground frame)
Z

t

Z
uθ̇w dt = K.E.(t) +

0

0

15

t

k φ̇2s dt

(27)

where K.E.(t) is the total kinetic energy of the system. The second term on the right hand side of
represents the cumulative energy lost due to damping losses up to time t. The energy lost in damping is
precisely equal to the work done by the damping force, that is
Z

φs (t)

L.E.(t) =

Z
k φ̇s dφs =

0

t

k φ̇2s dt.

(28)

0

The term on the left hand side of Eq. (27) represents the total energy pumped into the system by
the motor, up to time t. The power input into the system by the motor is the sum of power imparted to
the wheel by the actuation force u(t), and the power imparted to the stool by the reaction force. The
cumulative input energy (denoted by I.E.(t)) is given by
Z

t

Z

t

Z

(29)

0

0

0

t

uθ̇w dt

(−u)φ̇s dt =

u(θ̇w + φ̇s ) dt +

I.E.(t) =

We can simplify the energy balance (27) as

I.E.(t) = K.E.(t) + L.E.(t)

(30)

Energy description for the infinite-dimensional fluid system: The above energy equation can be
extended for the infinite dimensional case as follows.
1
I.E.(t) = K.E.(t) + ρ
2

Z

1
v dV + ρν
2
2

Z tZ 
0

∂vi
∂vk
+
∂xk
∂xi

2
dV dt

(31)

where the second term on the right hand side is the kinetic energy of the fluid, while the third term
is the energy lost due to viscous damping [8]. The velocity of the fluid in cartesian coordinates is given
by v = −v sinθ î + v cosθ ĵ, where v is the tangential velocity of the fluid as discussed previously
(4). We consider the system at a state when the wheel is no longer being actuated and is in a state
of rest with respect to the stool. The system can be found in such a state at the extreme position of
an oscillation that occurs after the wheel has been brought to rest. This implies that I.E(t) in (31)
is zero and hence, the kinetic energy of the stool can only be exchanged with that of the fluid, or be
lost due to damping. We analyze this state with the parameters from table 1 for the fluid-stool-wheel
system. At the extremum of the oscillation, the stool-wheel system is momentarily at rest, and as is
clear from (31), must be imparted energy from the fluid to start rotating again. The maximum energy
that the stool-wheel system could possibly regain is therefore equal to the kinetic energy of the fluid at
this extreme position. This leads to an expression (32) for the minimum fluid velocity (averaged over
the thickness of the annulus) in terms of system constants and the velocity of the stool.
 i 2
1h 
1
(Iw + Is )φ̇2s = ρπ (Ri + t)2 − (Ri )2 h vavg
2
2

(32)

where t and h are the thickness and height of the bearing respectively, while vavg is the velocity of the
fluid averaged over the thickness of the annulus. Even for a stool velocity as low as 1 rpm, the required
average fluid velocity is around 11 rpm, which is quite high and unrealistic. Hence, it is likely that a

16

different model is required to address such behaviour.
Name
Moment of Inertia of Wheel-Stool System
Inner diameter of bearing
Thickness of bearing
Height of bearing
Density of Fluid

Value
1
0.1
0.01
0.01
1000

Units
kg/m2
m
m
m
kg/m3

Table 1: Parameter values of fluid-wheel-stool system that are representative of experiment in [5]

8

Conclusion

In this paper, we present certain aspects of the damping-induced self recovery phenomenon that have
not been investigated so far in the existing literature. We present a technique to reduce the infinitedimensional fluid model to the better understood finite dimensional case, by deriving a formula for an
effective damping constant. We show that a bifurcation takes place at k = 0, and upon varying the value
of k from zero to a positive number, the stability of the stool changes from neutral to asymptotically
stable. We also derive an expression for the angle at which boundedness occurs to validate the approximation of the fluid system (by comparing with numerical experiments). Finally, we present an energy
description of the system to give an intuitive understanding of the energy dynamics, and to point out
that further experimental and theoretical investigation is necessary to explain the peculiar oscillations
found in experiments [5, 6].

References
[1] Dong Eui Chang and Soo Jeon. Damping-induced self recovery phenomenon in mechanical systems with an unactuated cyclic variable. Journal of Dynamic Systems, Measurement, and Control,
135(2):021011, 2013.
[2] Dong Eui Chang and Soo Jeon. On the damping-induced self-recovery phenomenon in mechanical
systems with several unactuated cyclic variables. Journal of Nonlinear Science, 23(6):1023–1038,
2013.
[3] Dong Eui Chang and Soo Jeon. On the self-recovery phenomenon in the process of diffusion. arXiv
preprint arXiv:1305.6658, 2013.
[4] Dong Eui Chang and Soo Jeon. On the self-recovery phenomenon for a cylindrical rigid body
rotating in an incompressible viscous fluid. Journal of Dynamic Systems, Measurement, and Control,
137(2):021005, 2015.
[5] Dong Eui Chang and Soo Jeon. Video of an experiment that demonstrates the self-recovery phenomenon in the bicycle wheel and rotating stool system. https://youtu.be/og5h4QoqIFs.

17

[6] Andy
bikes,

Ruina.
arrows,

Dynamic
and

walking

muscle-smarts:

2010:

Cats,

astronauts,

Stability,

translation,

and

trucks,
rotation.

http://techtv.mit.edu/collections/locomotion:1216/videos/8007-dynamic-walking-2010-andyruina-cats-astronauts-trucks-bikes-arrows-and-muscle-smarts-stability-trans, 2010.
[7] Dong Eui Chang and Soo Jeon. Video of an experiment that demonstrates the self-recovery phenomenon in the vessel and fluid system. https://youtu.be/26qGQccK4Rc.
[8] L. D. Landau and E. M. Lifshitz. Fluid Mechanics, volume 6. 1959.
[9] Stanley J Farlow. Partial differential equations for scientists and engineers. Courier Corporation,
1993.
[10] Milton Abramowitz and Irene A Stegun. Handbook of mathematical functions: with formulas, graphs,
and mathematical tables, volume 55. Courier Corporation, 1964.
[11] S.H. Strogatz. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry,
and Engineering. Advanced book program. Avalon Publishing, 1994.

Appendix A

Validation of effective damping constant

With the computed, effective damping constant kef f (18) of the infinite-dimensional fluid system (4),
we find the angle at which boundedness is attained, given by
φ∗s =

−Iw vw (Ro − Ri )
2πρνRo Ri2

(33)

We now present some numerical experiments by solving the PDE-system (4) using the method of
lines. Equation (18) is then verified by comparing the numerical values obtained using formula (33)
and the numerical solutions of the original PDE. It is observed that the error tends to zero as Ro /Ri
approaches 1, independent of the value of Ri . The material constants used for the simulation, and for
the theoretical calculation are tabulated below

18

Ri (cm)

Ro (cm)

13.5
13.5
13.5
13.5
13.5
13.5
13.5
13.5
13.5

13.51
13.68
13.75
14
14.5
15
15.5
20
27

27
27
27
27
27
27
27
27
27

27.02
27.36
27.5
28
29
30
31
40
54

0.07
1.33
1.85
3.70
7.41
11.11
14.81
48.15
100

With the PDE
Angle (rad)
6.16
108.7
149.8
291.7
553.8
790.3
1004
2265
3123

With kef f
Angle (rad)
6.16
109.5
151.3
297.1
573.7
831.9
1073
2704
4160

0.07
1.33
1.85
3.70
7.41
11.11
14.81
48.15
100

1.54
27.19
37.47
72.96
138.5
197.6
251.1
566.3
780.8

1.54
27.37
37.81
74.28
143.4
208
268.4
675.9
1039.9

(Ro −Ri )
Ri

× 100

% Error
0.00
0.74
1.00
1.85
3.59
5.26
6.87
19.38
33.21
0.00
0.66
0.91
1.81
3.54
5.26
6.89
19.35
33.18

Table 2: A comparison of the angles at which boundedness occurs, i.e., between the numerical simulations
and the theoretically derived formula (33), for different values of (Ro − Ri )/Ri .

Name
Moment of Inertia of Wheel
Moment of Inertia of Stool
Kinematic Viscosity of Fluid
Density of Fluid
Steady State Velocity of Wheel
Proportional Control Parameter
Derivative Control Parameter

Value
6 × 10−3
1.96
1.17 × 10−6
1.0147 × 103
60π
1
100

Units
kg/m2
kg/m2
m2 /s
kg/m3
rad/s
-

Table 3: Parameter constants used for the simulations.

19

