Proceedings of the ASME 2015 International Mechanical Engineering & Exposition
IMECE2015
November 13-19, 2015, Houston, Texas, USA
DRAFT IMECE2015-52374
A THEORETICAL AND NUMERICAL STUDY OF THE DZHANIBEKOV AND TENNIS
RACKET PHENOMENA
Hidenori Murakami
Department of Mechanical and Aerospace
Engineering
University of California, San Diego
La Jolla, California, USA
Oscar Rios
Department of Mechanical and Aerospace
Engineering
University of California, San Diego
La Jolla, California, USA
Thomas J. Impelluso
Department of Mechanical and
Marine Engineering
Bergen University College
Bergen, Norway
ABSTRACT
In this paper, we present complete explanation of the
Dzhanibekov phenomenon demonstrated in a space station
(www.youtube.com/watch?v= L2o9eBl_Gzw) and the tennis
racket phenomenon (www.youtube.com/watch?v=4dqCQqIGis). These phenomena are described by Euler’s equation of an
unconstrained rigid body that has three distinct values of
moments of inertia. In the two phenomena, the rotations of a
body about the principal axes that correspond to the largest and
the smallest moments of inertia are stable. However, the
rotation about the axis corresponding to the intermediate
principal moment of inertia becomes unstable, leading to the
unexpected rotations that are the basis of the phenomena. If this
unexpected rotation is not explained from a complete
perspective which accounts for the relevant physical and
mathematical aspects, one might misconstrue the phenomena as
a violation of the conservation of angular momenta. To address
this, especially for students, we investigate the phenomena
using more precise mathematical and graphical tools than those
employed previously.
Following Élie Cartan [1], we explicitly write the vector
basis of a body-attached, moving coordinate system. Using this
moving frame method, we describe the Newton and Euler
equations. The adoption of the moving coordinate frame
expresses the rotation of the body more clearly and allows us to
use the Lie group theory of special orthogonal group SO(3).
We integrate the torque-free Euler equation using the
fourth-order Runge-Kutta method. Then we apply a recovery
equation to obtain the rotation matrix for the body. By
combining the geometrical solutions with numerical
simulations, we demonstrate that the unexpected rotations
observed in the Dzhanibekov and the tennis racket experiments
preserve the conservation of angular momentum.
INTRODUCTION
In the 18th century, Leonhard Euler (1707-1783) developed
rigid-body dynamics. The equation of rotational motion of a
rigid body is referred to as Euler’s equation. As an extension
relevant to this paper, he also derived the equation that
described the dynamics of rigid bodies in torque-free motion
[2].
Both the Dzhanibekov and the tennis racket phenomena
occur during torque-free rotation of a rigid-body with three
distinct, principal moment-of-inertia. Thus, it is reasonable to
resolve this problem while also presenting the moving frame
method in dynamics. To consolidate previous contributions
related to the two phenomena, it is first necessary to define the
Newton and the Euler equations of motion for a rigid body
together.
Equations of motion of a Rigid Body
To locate points within a rigid body of mass m, we
establish a body-attached principal coordinate system
 s1 s2 s3  with its origin at the body’s center of mass, point
C. We define the unit vector e i (t ) tangent to the si -axis,
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e(t ) 
.
 e1 (t ) e 2 (t ) e3 (t )
Following Élie Cartan [1], we write the vector basis
explicitly to represent vectors. We do this to identify the
coordinate system that supports the vector components. To
facilitate this, we adopt Frankel’s compact notation [3], where
the vector basis e(t )   e1 (t ) e 2 (t ) e3 (t ) is expressed by a
i  1, 2, 3 , and from them form a vector basis
1 3 row matrix so that vector components are expressed by
3 1 column matrices [4].

rC (t )  e xC (t )  e
I
I
1
e
 x1C (t ) 

.
e  x2 C (t ) 
 x (t ) 
 3C 
I
2
I
3

(1)
We take the time differentiation, denoted by a superposed dot,
of Eq. (1) to obtain the velocity of the center of mass:
 x1C (t ) 

.
(2)
I
I
I
I


v C (t )  rC (t )  e xC (t )   e1 e 2 e 3  x 2 C (t ) 
 x (t ) 
 3C 
Let the total mass of the body be denoted by m, Newton’s
equation of a rigid body subjected to the gravitational force in
the  e 3I direction is written in vector form as:
(3a)
m v C (t )  mg e3I ,
and in components
 x1C (t ) 
 0 



.
I


e m  x2 C (t )   e m  0 
 x (t ) 
 g 


 3C 
I
(3b)
For the Dzhanibekov experiments in a space station, g  0 ,
while for the tennis racket experiments performed on the
ground, g  9.81 m/s 2 .
inertial coordinate system is defined as: e I   e1I e 2I e3I  .
For analytical simplicity, we select the inertial coordinate
system to be the body-attached coordinate system at time t=0,
e I  e(0) .
Finally, as the last critical point, the body attached
coordinates will be defined so that the mass moment of inertia
si -axis becomes J i C and the inequality
with the
Euler’s Equation for Torque Free Motion
The current body-attached vector-basis or frame, e(t ) , is
obtained from the inertial frame e I  e(0) by rotating it by R(t ) ,
which is a 3 3 rotation matrix with determinant one.
(4)
e(t )  e I R(t ) .
As with all rotation matrices, which form a Lie group of the
special orthogonal group, SO(3), the inverse is the transpose,
denoted with the superscript ‘T’;
(5)
e I  e(t ) ( R(t ))T .
We take the time derivative of Eq. (4) using Eq. (5) and obtain:
(6)
e (t )  e I R (t )  e(t ) ( R(t ))T R (t ) .
T 
From the last matrix product, ( R(t )) R(t ) , which is readily
shown to be a 3 3 skew symmetric matrix. Observing a oneto-one correspondence between 3 3 skew symmetric matrices
and three vector components, we define the skew symmetric
angular velocity matrix  (t ) [4]:
J1C  J 2 C  J 3C  0 is satisfied. In addition, we reemphasize in
e (t )  e(t )  (t ) ,
Fig.1: A body-attached principal-coordinate system
 s1 s2 s3 and an inertial coordinate system
 x1
x2
x3 
To define position vectors of the body, a fixed inertial
coordinate system  x1 x2 x3  with the origin at point O is
also introduced, as shown in Fig. 1. The unit coordinate vector
along the xi -axis is denoted by e i I . The vector basis for the
this paper that xi is reserved to designate the inertial coordinate
system, while
where

0
 (t )  ( R(t ))T R (t )   3 (t )
 2 (t )
si is reserved for the body-attached coordinate
system.
Newton’s Equation of Motion of the Center of Mass
In Fig. 1, the position vector of the center of mass, C, of the
body-attached frame is expressed, using the inertial frame, as:
(7a)
 3 (t ) 2 (t ) 
. (7b)
0
 1 (t )
1 (t )
0 
The corresponding angular velocity vector
ω(t ) becomes
 1 (t ) 

.
ω(t )  e(t )  (t )   e1 (t ) e 2 (t ) e 3 (t )  2 (t ) 
  (t ) 
 3 
(8)
Equations (7b) and (8) express one-to-one correspondence
between the skew symmetric matrices in the Lie algebra of
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SO(3), denoted by so(3), and vectors. In physics, skew
symmetric rank-2 tensors or two forms have corresponding
vector representation through the interior product of the vector
with the volume form [3].
Let the angular momentum vector about the center of mass
be denoted by H C (t ) :
 H1C (t ) 

.
H C (t )  e(t ) H C (t )  e(t ) H 2 C (t ) 
 H (t ) 
 3C 
(9a)
Its components, H C (t ) , with respect to the body attached frame
is expressed by the product between the principal moment of
inertia matrix J C and the components of the angular velocity
vector as:
 J 1C
0
0  1 (t ) 
 . (9b)


H C (t )  J C  (t )   0 J 2 C
0  2 (t ) 
 0
0 J 3C  3 (t ) 

Euler’s equation of rotational motion of a rigid body
without external torque is expressed in vector form as:
 (t )  0 .
(10)
H
C
Using the pivotal equation of the moving frame method, Eq.
(7a), the time derivative of the angular velocity vector is
computed as:
 (t )  d (e(t ) H (t ))  e (t ) H (t )  e(t ) H (t )
H
C
C
C
C
dt
 e(t ) H C (t )   (t ) H C (t ) .
Thus, Euler’s equation with respect to the body attached frame,
using Eq. (9b), becomes:
(11a)
e(t ) H C (t )   (t ) H C (t )  0 ,




and using Eq. (9b):
e(t ) J C (t )   (t ) J C (t )  0 .


(11b)
We first note that e(t ) (t )H C (t )  ω(t )  H C (t ) , justifying a
new notation that avoid the complexity of the cross product.
Second, if the moving frame e(t ) is not written explicitly, as Eq.
(11a), the term which corresponds to e(t ) H C (t ) must be
expressed by introducing another time-differential operator that
works only on the vector components with respect to the bodyattached vector basis, e(t ) . Representative symbols used for
 (t )
[5-8].
e(t ) H (t ) are H (t ) / t , d ' H (t ) / dt , and H
C
C
C
C
s cood
Returning to Eq. (11a), to find the components expression,
we substitute Eqs.(7b) and (9b) into Eq.(11):
 H 1C (t )   0
 3 (t ) 2 (t )  H1C (t )   0 

 

  

0
 1 (t ) H 2 C (t )    0  ,(12a)
 H 2 C (t )    3 (t )
 H (t )    (t )  (t )
0  H 3C (t )   0 
1
 3C   2
 J1 C
0
0  1 (t ) 



0   2 (t ) 
 0 J2C
 0
0 J 3 C   3 (t ) 

 3 (t ) 2 (t )   J1C
 0


  3 (t )
0
 1 (t )  0
 2 (t ) 1 (t )
0   0
0
J2C
0
0  1 (t )   0 
  

0  2 (t )    0  .
J 3 C  3 (t )   0 
(12b)
Equation (12b) yields Euler’s equation for a torque free motion:
(13a)
e1 (t ) : J1C1 (t )  ( J 2 C  J 3C )2 (t )  3(t )  0 ,
e 2 (t ) : J 2 C 2 (t )  ( J 3C  J1C )3 (t ) 1(t )  0 ,
(13b)
(13c)
e3 (t ) : J 3 C3 (t )  ( J1C  J 2 C )1 (t )  2(t )  0 .
The rotational motion of a body in the Dzhanibekov and the
tennis racket phenomena are described by Eqs. (13a-c).
We now present available analytical and geometrical
analyses of the equations that are relevant to the physical
interpretation of the phenomena.
Euler’s Analytical Solution of Torque-Free Rotations
In 1765, Euler [2] postulated and confirmed that four
quantities concerning a body’s motion were conserved: the
kinetic energy and the three components of angular momentum
vector expressed with respect to a fixed inertial coordinate
system. We now derive these conserved quantities from Eq.
(10) and (11).
Let the rotational kinetic energy be defined by K rot :
1
1
(14)
K rot (t )  ω(t )  H C (t )  ( (t ))T J C (t )  .
2
2
We take the time derivative of Eq. (14) using J C  ( J C )T :


1
K rot (t )  ( (t ))T ( J C )T  (t )  ( (t ))T J C  (t )  ( (t ))T J C (t )  .
2
Then we apply Eq. (11b) in situations without external torque to
obtain
K rot (t )  ( (t ))T  (t )J C (t ) .
Finally, we obtain the conservation of the rotational energy:
(15)
K rot (t )  0 ,
by noting the vanishing right-hand side, either by performing
the multiplications of the right hand side of Eq. (15) or
observing the equivalency of the right-hand-side to the scalar
triple product:  ω(t )  ω(t )  HC (t ).
We combine Eqs. (14) and (15) using the initial rotational
kinetic energy, K rot (0) , to express the conservation of rotational
kinetic energy. The resulting equation then expresses an
ellipsoid with respect to the 1 ,  2 , 3 -axes:
J1C (1 (t )) 2  J 2 C (2 (t )) 2  J 3C (3 (t )) 2  2K rot (0) , (16a)
and an ellipsoid with respect to the H1C , H 2 C , H 3C -axes [9]:
( H1C (t )) 2 / J1C  ( H 2 C (t )) 2 / J 2 C  ( H 3C (t )) 2 / J 3C  2K rot (0) .
(16b)
Both ellipsoids are referred to as energy ellipsoids or inertia
ellipsoids.
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Euler’s second set of conserved quantities: the components
of angular momentum with an inertial coordinate frame are
found from Eq. (10)
(17)
H C (t )  H C (0) .
We express Eq. (17) with the inertial coordinate frame as:
(18)
e I H CI (t )  e I H CI (0) .
The following issue is of significance. When the body
rotates, e (t )  0 , the conservation of angular momentum, Eq.
(17), does not enforce that the components of the angular
momentum vector decomposed with the body-attached moving
frame e(t ) remain constants. They can change according to Eq.
(12a), while satisfying Eqs. (10) and (17). It is this
understanding which justifies the power of the moving frame
method which can extract equations in any relevant frame.
Euler also noted that the length H C (t ) of the angular
momentum vector remains the length at t=0 from Eqs. (10) or
(17), [9]:
2
(19a)
( J1C1 (t )) 2  ( J 2 C2 (t )) 2  ( J 3C3 (t )) 2  H C (0) .
2
( H1C (t )) 2  ( H 2 C (t )) 2  ( H 3C (t )) 2  H C (0) ,
(19b)
Geometrically, for a constant-length, angular-momentum
vector, Eq. (19a) with respect to the 1 ,  2 , 3 -axes forms an
ellipsoid , while Eq. (19b) with respect to the H1C , H 2 C , H 3C axes forms a sphere. They are referred to as the angular
momentum ellipsoid and the angular momentum sphere,
respectively.
Using Eqs. (16a) and (19a), Euler in 1758 [2] obtained an
analytical solution for  (t ) of Eqs. (13a-c). His solution was
later rewritten by using Jacobi’s elliptic functions, which was
introduced by Carl Jacobi (1804-1851) in 1829. Wittenburg in
1977 concisely presented Euler’s solution using Jacobi’s elliptic
functions in his monograph [10, 11].
Euler observed from his equations that three steady-state
solutions referred to as permanent rotation states exit:
(20a)
1 (t )  ˆ 0 , 2 (t )  3 (t )  0 ,
1 (t )  0 , 2 (t )  ˆ 0 , 3 (t )  0 ,
1 (t )  2 (t )  0 , 3 (t )  ˆ 0 ,
(20b)
(20c)
where ̂ 0 is a constant.
In certain cases, when a dominant angular velocity is
applied about one principal axis to induce a state of permanent
rotation, the initial angular velocity deviates slightly from the
state of permanent rotation. In other words, in addition to the
dominant component about the principal axis, the initial angular
velocity includes small angular velocity components about other
axes.
To account for the slight perturbation of initial
conditions, the stability of the angular velocity components near
the permanent rotation states must be examined.
A majority of historians who investigated Euler’s work
concluded that he also observed the following. When a body
was given an initial spin with respect to the principal axes with
the maximum and minimum moments of inertia, the rotations
remained stable. However, when the initial spin was given with
the intermediate principal axis, the rotation became unstable.
Poinsot’s Geometrical Presentation of Euler’s
Solution: Polhodes
Fig. 2: Polhodes on a constant rotational energy ellipsoid
Louis Poinsot (1777-1859) added a geometrical
interpretation to Euler’s analytical solution of a torque-free
motion [12]. The angular velocity vector must lie on both the
constant energy ellipsoid, Eq. (16a), and the constant angular
momentum ellipsoid, Eq. (19a), ensuring that the two ellipsoids
intersect at least at one point. Therefore, the trajectory of the
angular velocity vector with respect to the body-attached
coordinate frame must lie on a path at the intersections of the
two ellipsoids. Each intersection, plotted on the energy ellipsoid
represents the trajectory of the angular velocity vector, as
illustrated in Fig. 2. For a given initial kinetic energy, K rot (0) ,
by changing the length of the initial angular momentum vector,
H C (0) , i.e, the size of the angular momentum ellipsoid,
various intersections are obtained. These intersections are
collectively referred to as polhodes by Poinsot [12].
Polhodes are closed curves except at the end points of three
principal axes of the energy ellipsoid. The end points of the
minor axis of the ellipsoid represents the permanent rotation
with the s1 -axis, corresponding to the maximum moment
inertia, J 1 . The end points of the major axis represents the
permanent rotation with respect to the s3 -axis, corresponding to
the minimum moment of inertia, J 3 . These end points of the
minor and major axes of the energy ellipse are centers, which
are surrounded by elliptic trajectories, as shown in Fig. 2.
Closed elliptic trajectories of the angular velocity components
indicate that the perturbed rotation will not leave the permanent
rotation state. The rotational response is almost like the
permanent rotation state for small perturbations of initial
conditions.
On the contrary, the end points of the intermediate axis,
corresponding to the permanent rotation about the s 2 -axis with
the intermediate moment of inertia, J 2 , are saddle points or a
hyperbolic fixed points,
which are passed through by
separatrices. Near each saddle point, the trajectories are
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hyperbolas indicating that perturbed rotations are unstable [712]. Following the hyperbolic paths, the perturbed rotation
deviates a lot from the permanent rotation state.
principal axes. Geometrically, both a box and a dictionary are
represented by cuboids.
We attach a Cartesian coordinate system  s1 s2 s3  with
the origin at the center of mass, point C. We choose the
coordinate axes parallel to the edges to form principal
coordinate axes. We name the axes, si , i 1, 2, 3 , in the
decreasing order of principal moment of inertia: J1  J 2  J 3  0 ,
as shown in Fig. 4 (a). The edges of the cuboid are a  b  c , in
the s1 , s 2 , and s3 -directions, respectively. The actual values of
the principal moment of inertia are as follows:
J1 
m 2
m
m
(b  c 2 ) , J 2  (c 2  a 2 ) , and J 3  (a 2  b 2 ) . (21)
12
12
12
Fig. 3: The rolling of the energy ellipsoid on the invariable
plane
Poinsot also explained the body rotation with respect to the
inertial coordinate system with the origin at the center of mass,
point C. He plotted the angular momentum vector, H C , in the
inertial coordinate space, where H C remains constant, due to
Eq. (17). Then, at the tip of the angular momentum vector he
defined a plane normal to the vector, which is referred to as the
invariable plane. From Eq. (14), for a constant rotational
kinetic energy, the projection of the angular velocity vector
onto the angular momentum vector is constant:
ω(t )  H C  2 K rot (0) . Since the tip of ω(t ) moves on the energy
(a)
ellipsoid while the ellipsoid rotates in the inertial space, the
tangent plane on the ellipsoid at the tip of ω(t ) must be on the
invariable plane, as illustrated in Fig. 3. The loci of the tip of
the angular velocity vector on the invariable plane are called the
herpolhodes to distinguish them from polhodes on the energy
ellipsoid.
Now, we have completed the presentation of the theoretical
background of a torque-free rotation of a rigid body, which are
critical for the explanations of the Dzhanibekov and the tennis
racket phenomena.
The Dzhanibekov Phenomenon
In a space station in 1985, Vladimir Dzhanibekov, a
Russian astronaut, conducted experiments concerning the
rotation of a rigid body. For a rigid body with three distinct,
moments of inertia, he applied an initial spin with each
principal axis. His experiments in the space station were
recorded and are available at the following websites:
www.youtube.com/watch?v=L2o9eBl_Gzw and
www.youtube.com/watch?v =dL6Pt1O_gSE.
In the space station, the gravitational attraction is negligible
in Newton’s equation (3b). Therefore, the translational velocity
of a rigid body remains constant. Therefore, we focus on the
rotational motion of the body.
In one of Dzhanibekov’s experiments in the space station,
both a box and a dictionary were spun with respect to the three
(b)
(c)
Fig. 4: (a) A cuboid with an initial spin about the J1 -axis; (b)
with an initial spin about the J 2 -axis; and (c) an initial
spin about the J 3 -axis.
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If the initial spin, ̂ 0 , was applied to the cuboid’s principal
axis, e1 , with the maximum moment of inertia, and the
principal axis, e 3 , with the minimum moment of inertia, the
decreasing order as: J1  J 2  J 3  0 . The moments of inertia of
tennis rackets may be measured experimentally [13] since due
to the symmetry, the principal planes are easily found.
cuboid continued rotating with the initial spin-axis unchanged.
These spin behaviors can be easily interpreted as the result of
the conservation of angular momentum since the rotational axes
do not change with time.
If the cuboid was spun with respect to the principal axis,
e 2 , with the intermediate moment of inertia, the cuboid rotates
while its own initial spin axis also starts rotating. The rotation
of the initial spin axis is due to the growing component of either
1 or 3 . This rotational motion of the initial spin-axis was an
unexpected event. In another experiment on a wing nut,
Dzhanibekov observed that the unexpected rotation of the initial
spin-axis, e 2 , is unstable, but periodic.
If we understand that the initial spin also includes minute
perturbations, the unexpected rotational phenomenon, induced
by an initial spin applied to the intermediate principal axis is
easily explained by the hyperbolic orbits near the saddle points
at the ends of the intermediate principal axis of a constant
energy ellipsoid in Fig. 2. Furthermore, all the hyperbolic orbits
near the intermediate permanent rotation are closed surrounding
either the 1 - or the 3 -axis in Figure 2. The closure of the orbit
with either the 1 - or the 3 -axis implies that the initial spin
axis also rotates periodically with the axis. This is what
Dzhanibekov observed in his experiments.
If one recalls that the conservation of angular momentum,
Eq. (10), and the paragraph immediately following Eq. (18) do
not prohibit the rotation of the initial spin axis, Eq. (12a), the
hyperbolic orbits near the saddle points explain Dzhanibekov’s
experiments. This includes the unexpected rotations observed
when the initial spin was given with the intermediate axis.
Although polhodes are exact solutions, presented
geometrically, what they do not show is: (i) how fast or slowly,
each orbit is traced by the angular velocity vector and (ii) how
the body is rotating with respect to the inertial coordinate frame.
For this we overlay the power of geometry with that of
numerical methods which will follow the next section.
The Tennis Racket Phenomenon
The tennis racket phenomenon is popularly referred to as
the “tennis racket theorem” even if it does not meet the criterion
to be promoted to a physical or mathematical theorem.
To explain the rotation of the racket, we define the
principal coordinate system  s1 s2 s3  at the center of mass,
as illustrated in Figs. 5 (a) and (b). The vector basis e(t ) 
 e1 (t ) e2 (t ) e3 (t )is defined using the unit tangent vectors to
the coordinate axes. The s2 , s3 -plane defines the head plane
and the s1 -axis is normal to the head plane. With this coordinate
system, the principal mass moment of inertia appear in the
(a)
(b)
(c)
(d)
(e)
Fig. 5: (a) The head plane of a tennis racket, (b) a side view
normal to the head plane of the racket, (c) a case with
an initial spin with the normal axis to the head plane,
(d) a case with an initial spin with the axis normal to
the grip axis in the head plane, and (e) a case with an
initial spin with the grip axis.
To observe the rotational response, we toss a racket
vertically up, simultaneously giving a spin along one of the
principal axes of the racket. When we apply the initial spin with
the s1 -axis with the maximum moment of inertia, the racket
keep spinning in a stable manner. A similar stable rotation is
observed when the initial spin is applied with respect to the s3 axis with the minimum moment of inertia. However, when the
initial spin is applied with the s 2 -axis with the intermediate
moment of inertia, in addition to the rotation of the racket with
the s 2 -axis, the axis also rotates with the s3 -axis.
A recorded experiment is available at the web site:
www.youtube.com/watch?v=4dqCQqI-Gis.
Although both Dzhanibekov’s and the tennis racket
phenomena are rotations of a torque-free body, the tennis racket
6
Copyright © 2015 by ASME
experiment is ill-fated since it cannot demonstrate a periodic
motion of the racket with the intermediate principal axis, shown
in Fig. 5 (d). Due to gravity being a central force, the racket
undergoes free-torque rotation as well as the vertical motion.
The racket is up in the air only for a short duration. The time
from the tossing to the catching is 2v0 / g using Newton’s
equation (3b) if the initial vertical velocity given to the racket at
the center of mass is x0 (0)  v0 . Therefore, to compensate the
shortcoming of the experiment, Poinsot’s polhodes must be
presented to reveal the nature of the tennis racket phenomenon.
In 1991, Ashbaugh, Chicone, and Cushman [14] reported
the analytical solution for the tennis racket experiments by
solving Eq. (12a) with respect to the components of the angular
momentum vector. They only showed the final result and did
not show the intermediate derivations. However, it is expected
that their analysis parallels the analytical solution presented for
the angular velocity components by Wittenburg [10]. Both
Wittenburg and Ashbaugh et al. used the Euler angles to
express the rotation of the tennis racket by tactically avoiding
the critical point of the Euler angles.
Without showing the global orbits on the energy ellipsoid,
expressed with body axes, either with respect to the angular
velocity components, in Fig. 2, or the angular momentum
components, it is difficult to explain the conservation of angular
momentum regarding the strange rotation. The confusion could
grow further, if students are only presented the linear stability
analysis. This linearized analysis shows that the rotation,
induced initially by the pin with the intermediate axis,
exponentially grows. However, this does not indicate a globally
diverging rotation since the linear stability analysis only holds
near the permanent rotation state.
The Work Presented in This Paper
To summarize the background for the torque-free motion of
a rigid body, an analysis is readily available. The solution is
obtained by using the conservations of energy and angular
momentum. Furthermore, polhodes on the energy ellipsoid help
us understand the global rotational behavior. Although polhodes
in Fig. 2 are easy to visualize, however, it is difficult to
visualize the rotation of the body itself as well as the rotation of
the energy ellipsoid with the body-attached coordinate axes.
Although Figure 3 may be used to present the conservation of
angular momentum vector using the invariable plane, it is
necessary to show numerically that the angular momentum
vector components with the inertial frame remain constant.
In the following, we analyze one of Dzhanibekov
experiments on the motion of a wing nut as shown in the video:
http://www.youtube.com/watch?v=dL6Pt1O_gSE. In addition,
we analyze a tennis racket experiment. Our approach employs
numerical integration of Euler’s torque-free equation and the
recovery equation for rotation matrix to present three
dimensional (3D) animations. To present the accuracy of the
numerical solutions, we compare them with the orbits
analytically obtained on the energy ellipsoid. The numerical
analyses plotted on the energy ellipsoid supplement the missing
information of the geometrical solution: how quickly or slowly
each orbit is traced.
FORMULATION OF THE TORQUE-FREE ROTATION
OF A RIGID BODY
We first formulate a torque-free rotation of a body with
three distinct values of the moments of inertia. Then, we present
numerical integration scheme as well as the geometrical
solution for the angular momentum.
Euler’s Equation for a Torque-Free Body
We define the initial value problem of the angular
momentum vector expressed by the body-attached coordinate
frame, e(t ) , Eq. (9a). Euler’s equations (12a) can be rewritten
using the inverse of Eq. (9b) as follows:
1
1
e1 (t ) : H 1C (t )  (

) H 2 C (t ) H 3C (t )  0 ,
J 2 C J 3C
1
1
e2 (t ) : H 2 C (t )  (

) H 3 C (t ) H1C (t )  0 ,
J 3 C J1C
1
1
e3 (t ) : H 3 C (t )  (

) H1C (t ) H 2 C (t )  0 .
J1 C J 2 C
The initial conditions at t=0 are prescribed as:
 H1C (0)   J1C 1 (0)   Hˆ 1 

 
  ,
 H 2 C (0)    J 2 C 2 (0)    Hˆ 2 
 H (0)   J  (0)   Hˆ 
 3C   3C 3   3 
where Ĥ 1 , Ĥ 2 , and Ĥ 3 are constants.
(22a)
(22b)
(22c)
(23).
We will integrate Eqs. (22a-c) using the fourth-order
Runge-Kutta method for discrete times with time increment t .
For given values of H i C (t ) , Eqs. (22a-c) give H i C (t  t ) ,
For the known angular momentum vector
i  1, 2, 3 .
components, the angular velocity components  (t ) and
 (t  t ) can be computed from the known angular momentum
components using the inverse of Eq. (9b).
Next, we present the time integration of angular velocity
matrix to find the rotation matrix without using angular
coordinates.
Reconstruction of Rotation Matrix
When a rigid body freely rotates, any coordinate
representation using three angular-coordinates may experience
its critical point. To avoid critical points, four angularcoordinates, such as quaternion, are used to express rotation
matrices [15, 16]. However, the resulting equations of motion
become more complicated than the original Euler’s equations.
Here, we present a simpler method to deal with rotation matrix
directly without resorting to four angular coordinates. This is a
direct result of the Moving Frame Method in Dynamics. There
are no critical points where the representation of rotation fails.
7
Copyright © 2015 by ASME
Equation (7b) gives reconstruction formula for R(t ) [17,
4]:
(24)
R (t )  R(t )  (t ) .
When the angular velocity remains constant,  (t )  0 , Eq.
(24) can be integrated analytically.
The Case with Constant Angular Velocity and
Analytical Solution
The solution of Eq. (24) with the initial value R(0) becomes
[3]:
(25)
R(t )  R(0) exp(t 0 ) ,
where the matrix exponential is defined as:

t2
t3
tk
exp(t 0 )  I d  t 0  ( 0 ) 2  ( 0 )3     ( 0 ) k .
2!
3!
k 0 k!
(26)
In order for Eq. (25) to be useful, we need a simple
expression for the matrix exponential. In what follows, we
simplify Eq. (26) using the Cayley-Hamilton theorem [18].
To simplify the computation of ( 0 ) k in Eq. (25), we first
compute the characteristic equation of 0 :
 
det( 0   I d )  det  30
 20
 30

10
20 
2
 10   3   ω 0  0 ,
  
(27a)
where
ω0  (01)  (02 )  (03 ) .
From Eq. (27a), the Cayley-Hamilton theorem gives
2
( 0 )3   ω0 ( 0 ) .
2
2
2
(27b)
2
(28)
Using Eq. (28), the right hand side of Eq. (26) is expressed
by I d , the 3 3 identity matrix, 0 , and ( 0 ) 2 :
t2
t3
2
exp(t 0 )  I d  t 0  ( 0 )2  ω0 0
2!
3!
5
6
t4
t
t
2
4
4
 ω0 ( 0 )2  ω0 0  ω0 ( 0 )2  .
4!
5!
6!
Collecting each power of 0 , we obtain
exp(t 0 )  I d  0 { t 
 ( 0 ) 2 {
t3
ω0
3!
t2 t4

ω0
2! 4!
2

2

t5
ω0
5!
t6
ω0
6!
4

4

t7
ω0
7!
t8
ω0
8!
6
0
ω0
{t ω0 
Numerical Integration of the Reconstruction Formula
for Rotation Matrix
We can integrate Eq.(24) analytically from t to t  t , if
the angular velocity is constant. While this is not the case here,
it can be applied to one time step of the Runge-Kutta
integration. Therefore, we adopt the mid-point integration
method using the mean value of the angular velocity,
(t  t / 2) [19]:
(30)
(t  t / 2)  (t )  (t  t )/ 2 .
Assuming that the angular velocity is constant during the time
step from t to t  t , we integrate Eq. (24) analytically for the
initial value of R(t ) to find R(t  t ) :
(31)
R(t  t )  R(t ) exp{t  (t  t / 2)} ,
where Eq. (29) is used to evaluate the exponential matrix for
0   (t  t / 2) .
Geometrical Solutions
The exact analytical solution is evinced geometrically using
the two integrals of Euler’s equations, Eq. (22a-c): the
conservations of energy and the conservation of the length of
the angular momentum vector.
We use Eq. (16b) to construct the energy ellipsoid
expressed with respect to the H1C , H 2 C , and H 3C -axes as
follows.
Let the initial rotational kinetic energy be denoted by K 0 .
The energy ellipsoid is expressed as:
 H1C (t )

 2 K 0 J1C

2
   1
1
1
1
  0  { (t ω0 ) 2  (t ω0 ) 4  (t ω 0 ) 6  (t ω0 )8  }.
ω
2
!
4
!
6
!
8
!
0


2
  H 3C (t )
 
  2 K 0 J 3C
 
2

  1 . (32a)




The major semi-axis of the ellipsoid is
2 K 0 J1C along the body-
attached H1C -axis, the intermediate semi-axis is
the
1
1
1
(t ω0 )3  (t ω0 )5  (t ω0 )7  }
3!
5!
7!
2
  H 2 C (t )
 
  2K 0 J 2C
 
K 0  K rot (0)  ( Hˆ 1 ) 2 / J1C  ( Hˆ 2 ) 2 / J 2 C  ( Hˆ 3 ) 2 / J 3C / 2 . (32b)
 }
 }.
2

 (1  cos(t ω 0 ) . (29)


ω0 in the direction of
the angular velocity vector, Eq. (29) agrees with the Rodrigues
formula [10, 11]. The present linear algebraic analysis yields
the same result as the geometric analysis adopted by Rodrigues.
Equation (29) is used in the numerical integration of Eq.
(24) when the angular velocity changes with time.
where
6
exp(t 0 )  I d

 
exp(t 0 )  I d 
sin(t ω0 )   0
ω0
 ω0
If we define a unit vector u  ω0 /
0
body-attached
H 2C -axis
and
the
2 K 0 J 2C along
minor
semi-axis
2 K 0 J 3C along the body-attached H 3C -axis.
We use Eq. (19b) to construct the angular momentum
sphere:
(33a)
( H1C (t )) 2  ( H 2 C (t )) 2  ( H 3C (t )) 2  2K 0 D ,
Finally, the simplified equation is found:
8
Copyright © 2015 by ASME
where 2 K 0 D represents the radius of the sphere as well as the
length of the angular momentum vector. Here, we have
parameterized the squared length of the angular momentum
vector by introducing a parameter D as:
2
2K 0 D  HC (0)  ( Hˆ 1 ) 2  ( Hˆ 2 ) 2  ( Hˆ 3 ) 2 , (33b)
where D has the dimension of the moment of inertia, J i C .
By changing the radius 2 K 0 D of the angular momentum
sphere through the parameter D, we can find the exact
trajectories of the angular momentum vector. This is done by
expressing the intersections between the energy ellipsoid, Eq.
(32a), and the angular momentum spheres of various radii, Eq.
(33a) in the body-attached frame.
Before continuing, we remind the reader that we will
leverage the following inequality: J1C  J 2 C  J 3C  0 . For a
sphere to intersect with the energy ellipsoid, its radius
2 K 0 D must be less than or equal to the major semi-axis of the
ellipsoid and greater than or equal to the minor semi-axis. This
requirement imposes the inequality for the parameter D:
(34)
J 3C  D  J1C .
The intersection of the ellipsoid and the sphere shows the exact
trajectories of the angular momentum vector expressed with the
body-attached coordinate frame.
To plot a constant energy ellipsoid and intersecting spheres
of various radii, i.e., different lengths of the angular momentum
vector, we first non-dimensionalize Eqs. (32a) and (33b)
using J 2 C .
The
non-dimensional
angular momentum
components and the ratio of moments of inertia are shown by
over-bars:
(35a)
H i C  H i C / 2K 0 J 2 C , i 1, 2, 3 ,
d  D / J 2C .
(35b)
The non-dimensional energy ellipsoid becomes, from Eqs. (32a)
and (35):
 J 2C

J
 1C

J
H1C 2  H 2 C 2   2 C

J

 3C
The major semi-axis of the ellipsoid is

H 3C 2  1 .


We first consider the projection on to the H 2 C , H 3C -plane.
The equation for the trajectory is obtained by eliminating
H1 C from Eqs. (36) and (37a):






1  J 2C H 2 C 2   J 2C  J 2C H 3C 2  1  J 2C d   0 , (38)


J



J
J
J
1C 
1C 
1C

 3C


where the inequality on the right –hand side is obtained by
using the inequality (37b). The coefficients on the left hand side
are both positive. Therefore, Eq. (38) shows an ellipse if the
right-hand-side is positive, else it shows the state of permanent
rotation: H1C  d , H 2 C  H 3C  0 . These elliptic trajectories
appear inside the projected energy at H1 C  0 in Eq. (36).
Second, we consider the projection on to the H 1C , H 2 C plane. The equation for the trajectory is obtained by eliminating
H 3 C from Eqs. (36) and (37a):
 J 2C J 2C 





H1C 2   J 2C  1H 2 C 2   J 2C d  1  0 , (39)

J





 3C J1C 
 J 3C

 J 3C

where the inequality on the right –hand side is obtained by
using the inequality (37b). Since the coefficients on the left
hand side are both positive, Eq. (39) represents an ellipse if the
right-hand side does not vanish, d  J 3C / J 2 C , i.e., D  J 3C . If
the right-hand side of Eq. (40) vanish, d  J 3C / J 2 C , i.e.,
D  J 3C , it becomes a point showing the state of permanent
rotation: H1C  H 2 C  0 and H 3 C  d . The projection of the
energy ellipsoid becomes an ellipse at H 3 C  0 in Eq. (36).
Finally, we consider the projection on to the H 1C , H 3C plane. The projected ellipsoid becomes an ellipse at H 2 C  0 in
Eq. (36). The equation for the trajectory is obtained by
eliminating H 2 C from Eqs. (36) and (37a):
 J
 1  2C

J1C

(36)
J1C / J 2 C along the
body-attached H1C -axis, the intermediate semi-axis is one
along the body-attached H 2C -axis and the minor semi-axis
J 3C / J 2 C along the body-attached H 3C -axis.
The non-dimensional angular momentum sphere is from Eqs.
(33b) and (35):
(37a)
H1C 2  H 2C 2  H 3C 2  d ,
where the radius d of the sphere satisfies the non-dimensional
inequality imposed by Eq. (34):
J 3C
J 1C
.
(37b)
d 
J 2C
J 2C



H1C 2   J 2C  1H 3C 2  1  d ,

J


 3C

(40a)
where the coefficients in pairs of parentheses on the left hand
side are both positive. Therefore, Eq. (38) shows a hyperbola.
The equations of its asymptotes are
H 3C  
J 3C ( J1C  J 2 C )
J1C ( J 2 C  J 3C )
H 2C .
(40b)
If d  1 , the vertices and focal points of the hyperbola are on
the H 3C -axis in the projected plane. Therefore, on the energy
ellipsoid, the trajectory is closed and circulates about the H 3C axis. On the contrary, if d  1 , the vertices and focal points of
the hyperbola are on the H 1C -axis in the projected plane.
Therefore, on the energy ellipsoid, the trajectory is closed and
circulates about the H1C -axis.
When d  1 , i.e., D  J 2 C , the trajectory of Eq. (39a) also
becomes Eq. (39b). Therefore, Eq. (39b) becomes separatrices.
9
Copyright © 2015 by ASME
J 3C / J 2 C  0.5 . For d  0.65, 0.8, 0.95, 1.05, 1.20, 1.35,
Furthermore, if H1C  H 3C  0 , it represent the state of
permanent rotation: H 2 C  1 .
1.5, 1.65, 1.8, 1.95, the projection of the trajectories: (b)
onto the H 2 C , H 3C -plane, (c) onto the H1C , H 3C -plane, and
Figure 6 (a) shows the intersecting energy ellipsoid and the
angular momentum sphere for J1 C / J 2 C  2 and J 3C / J 2 C  0.5 .
For d  0.65, 0.8, 0.95, 1.05, 1.20, 1.35, 1.5, 1.65, 1.8, and
1.95, the projected trajectories onto the H 2 C , H 3C -plane is
shown in Fig. 6 (b), those onto the H1C , H 3C -plane is in Fig. 6
(c), and those onto the H1C , H 2 C -plane in Fig. 6 (d)
The projection onto the
H1C , H 2C - plane
H 3C
The projection onto the
H 1C , H 3C - plane
(a)
H 1C
H 2C
The projection onto the
H 2C , H 3C - plane
As Fig. 6 (a) illustrates the trajectories that form a pair of
hyperbolas on the projected H1C , H 3C -plane is closed on the
energy ellipsoid, indicating that the rotation is periodic.
NUMERICAL RESULTS
We present the numerical simulations of Dzhanibekov’s
experiment of an unscrewing wing nut and the tennis racket
experiment. For numerical solutions, we integrate Euler’s
equations (22a-c) using the fourth-order Runge-Kutta method
and the recovery formula for the rotation matrix, Eq. (24), using
the mid-point method, Eq. (31) and Eq. (29).
Dzhanibekov’s experiment
For the numerical simulation, we used the following
properties for the mass moment of inertia of the wing nut:
2
2
J1C  3.036  106 kg/m , J 2 C  2.741  106 kg/m , and J 3C 
2
0.699  106 kg/m . The intermediate moment of inertia appears
in the axial direction of the nut, as illustrated in Fig. 7. The
ratios of the moments of inertia become J1C / J 2 C  1.11 and
H 3C
H 3C
(d) onto the plane H1C , H 2 C -plane.
J 3C / J 2 C  0.26 .
H 2C
(b)
Figure 7 shows a sequence of snap shots from the
numerical simulation, where an unscrewed wing nut advancing
to the left just before the separation from the bolt with the major
angular velocity of H 2 C (0)  2.1252  105 kg/m2 s and minute:
2
2
H1C (0)  0.001 kg/m s and H 3C (0)  0.001 kg/m s. The
H 3C
H 3C
computation of the rotational energy and the length of the
angular momentum vector gives d  D / J 2 C  1.0
d 1
d 1
(c)
H1C
d 1
H 2C
(d)
H1C
H1C
Fig. 6: (a) Angular momentum trajectories obtained from the
inter-sections between the energy ellipsoid and the
angular momentum sphere for J1C / J 2 C  2.0 and
Fig. 7: (a) A sequence of snap shots of the wing nut at times
t=0.00, 4.75, 5.25, 5.50 and 7.00 seconds
10
Copyright © 2015 by ASME
Fig. 7: (b) The angular momentum about the 1, 2, 3 axes and
the total angular momentum in the inertial frame
Figure 10 (a) A sequence of snap shots of the tenis racket.
Fig. 7: (c) The angular momentum about the 1, 2, 3 axes and the
total angular momentum in the moving frame
Fig. 8: (a) A sequence of snap shots of the wing nut: t= 0.0, 1.4,
1.8, 3.0, 4.0 seconds
The simulation in Fig. 7 (a), where the snap shots start from
the leftmost figure at t=4.75s, reproduces qualitatively the
periodic rotation of the wing nut observed by Dzhanibekov in
space.
Figure 7(b) presents the angular momentum about the
major spin axis and the total angular momentum (black lines).
The common value is 2.1252  105 kg - m 2 /s . The angular
momentum about the other two axes (blue) is 0.0.
Figure 7(c) presents the angular momentum about all three
axes in the moving frame, and the total angular momentum
(black line); the later retains the value 2.1252  105 kg - m 2 /s .
Fig 8: (b) The angular momentum about the 1, 2, 3 axes and the
total angular momentum in the inertial frame
The Tennis Racket Experiment
We present the simulation of a tennis racket, illustrated in
Fig. 5. The parameters are m=0.375 kg, J1C  0.0185 kg/m2,
2
2
J 2 C  0.0164 kg/m , J 3C  0.00121 kg/m .
We only present the result when the initial spin was applied
with the e 2 -axis in Fig. 5 (d). The initial angular velocities are:
2 (0)  5.0 rad/s, 1 (0)  3 (0)  0.001rad / s .
The vertical velocity of 24.5 m/s was applied at the center of
mass, which gives the racket air time of 5 seconds. The initial
value gives d  D / J 2 C  1.0
Fig. 8: (c) The angular momentum about the 1, 2, 3 axes and the
total angular momentum in the moving frame
Figure 8(b) presents the angular momentum about the major
spin axis and the total angular momentum (black lines). The
common value is 0.81kg - m 2 /s . The angular momentum about
the other two axes (blue) is 0.0.
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Figure 8(c) presents the angular momentum about all three
axes in the moving frame, and the total angular momentum
(black line); the later retains the value 0.81kg - m 2 /s .
The accuracy of the numerical solutions was checked by
plotting the angular momentum vector on the angular
momentum sphere, shown in Fig. 6 (a). This was done to
compare the momentum’s trajectory with the geometrically
exact trajectory. This geometrically exact trajectory is the
intersection between the energy ellipsoid and the angular
momentum sphere. For both simulations of the Dzhanibekov
and the tennis racket experiments, the numerically computed
trajectories of the angular momentum vector are
indistinguishably close to the exact orbit.
CONCLUDING REMARKS
For both the Dzhanibekov and the tennis racket experiments,
for torque free rotations, when the initial angular velocity is
applied to the axis of maximum or minimum moment of inertia,
the axis of rotation remains the same. This is easily interpreted
as the result of the conservation of angular momentum.
However, when the initial rotation is applied to the principal
axis of the intermediate value of moment of inertial, the body
exhibits the rotations about the other axes. This unstable
rotation could be easily misinterpreted as an evidence of the
violation of the conservation of angular momentum in torque
free motion.
In this paper, we theoretically and numerically demonstrated
(the latter with 3D animations), that the rotations are periodic
with the principal axis of the intermediate moment of inertia.
Our computational approach employs numerical integration of
Euler’s torque-free equation and a recovery equation for
rotation matrix to present three dimensional (3D) animations.
To present the accuracy of the numerical solutions, we
compared them with the orbits analytically obtained on the
energy ellipsoid.
NOMENCLATURE
H C (t ) : angular momentum vector
H C (t ) : magnitude of the angular momentum vector
e i (t ) , i  1,2,3 : unit tangent vectors to body attached
coordinate system
e iI , i  1,2,3 : unit tangent vectors to the inertial coordinate
system
e (t ) : time-rate of frame rotation
rC (t ) : the position vector of the center of mass, C, of the bodyattached frame
v C (t )  rC (t ) : the velocity vector of the center of mass, C, of
the body-attached frame
ω(t ) : angular velocity vector
D : arbitrary parameter with the dimension of the moment of
inertia
H i C (t ) :one components of the angular momentum vector
H C (t ) : coordinates of the angular momentum vector
H i C : the non-dimensional angular momentum
J i C : the mass moment of inertia with respect to the i-coordinate
K rot : rotational kinetic energy
R(t ) : a 3 3 rotation matrix with determinant one
R(t ) T : the transpose of a 3 3 rotation matrix
SO(3): the special orthogonal group
d  D / J 2 C : radius of non-dimensional angular momentum
sphere
g  9.81 m/s 2 : gravity and its constant value
m : mass of a body
so(3): the Lie algebra of SO(3)
 s1 s2 s3 : body-attached coordinate system.
t : time
t : time increment
I d : identity matrix
 x1
x2
x3 : an inertial coordinate system
 (t ) : angular velocity matrix
ACKNOWLEDGMENTS
The authors wish to thank Professor Emeritus Theodore
Frankel at the University of California, San Diego for his
continuous guidance and advice on the application of the Lie
group theory and Cartan’s moving frame method.
REFERENCES
[1] Cartan, E., 1986, On Manifolds with an Affine Connection
and the Theory of General Relativity, translated by A. Magnon
and A. Ashtekar, Napoli, Italy, Bibiliopolis.
[2] Euler, L., 1765, “Du mouvement de rotation des corps
solides autour d'un axe variable,” in Mémoires de l'académie
des sciences de Berlin 14, pp. 154-193; also in Opera Omnia:
Series 2, Volume 8, pp. 200 – 235.
[3] Frankel, T., 2012, The Geometry of Physics, an
Introduction, Third edition, Cambridge, New York.
[4] Murakami, H., 2013, A Moving Frame Method for MultiBody Dynamics, Proceedings of the ASME 2013 International
Mechanical Engineering Congress & Exposition, IMECE201362833, November 13-21, San Diego, California.
[5] Synge, J. L., 1960, Classical Dynamics, in Encyclopedia of
Physics, ed. S. Flügge, Volume III/1 Principles of Classical
Mechanics and Field Theory, Springer-Verlag, Berlin.
[6] Landau, L. D., and Lifshitz, E. M., 1960, Mechanics,
translated from Russian by J.B. Sykes and J.S. Bell, Pergamon
Press, NewYork.
[7] Goldstein, H., 1980, Classical Mechanics, second edition,
Addison-Wesley Publishing Company, Reading, Massachusetts.
[8] Greenwood, D. T., 1965, Principle of Dynamics, PrenticeHall, Inc., Englewood Cliffs, NJ.
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[9] Arnold, V. I., 1989, Mathematical Methods of Classical
Mechanics, second edition, Springer-Verlag, New York.
[10] Wittenburg, J., 1977, Dynamics of Rigid Bodies, B. G.
Teubner, Stuttgart, Germany.
[11] Wittenburg, J., 2008, Dynamics of Multibody Systems,
second edition, Springer, Berlin.
[12] Poinsot, L.,1834, Theorie Nouvelle de la Rotation des
Corps, Paris.
[13] Brody, H., 1985, “The moment of inertia of a tennis
racket,” The Physics Teacher, April, pp. 213-216.
[14] Ashbaugh, M., Chicone, C. C., and Cushman, A. H., 1991,
The Twisting Tennis Racket,” Journal of Dynamics and
Differential Equations, 3, pp. 67-85.
[15] Haug, E. J., 1989, Computer-Aided Kinematics and
Dynamics of Mechanical Systems, Vol. I: Basic Methods, Allyn
and Bacon, Boston.
[16] Nikravesh, P., 1988, Computer-Aided Analysis of
Mechanical Systems, Prentice Hall, Englewood Cliffs, New
Jersey.
[17] Holm, D. D., 2008, Geometric Mechanics, Part II:
Rotating, Translating, and Rolling, Imperial College Press,
London.
[18] Noble, B., and Daniel, J. W., 1977, Applied Linear
Algebra, 2nd edition, Prentice-Hall, Englewood, NJ.
[19] Hughes, T. J. R., 1987, The Finite Element Method, Linear
Static and Dynamics Finite Element Analysis, Prentice-Hall,
Englewood Cliffs, NJ.
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