Journal of Sound and Vibration (1998) 214(5), 833–851

Article No. sv981563

APPLICATIO N O F THE ABSO LUTE NO DAL

CO -O R DINATE FO R MULATIO N TO
MULTIBO DY SYSTEM DYNAMICS
J. L. E
Department of Mechanical Engineering, University of Seville, Av. Reina Mercedes s/n,
41012 Seville, Spain
H. A. H  A. A. S
Department of Mechanical Engineering, University of Illinois at Chicago,
842 West Taylor St., Chicago, IL 60607-7022, U.S.A.

(Received 30 May 1997, and in final form 4 February 1998)

The floating frame of reference formulation is currently the most widely used approach
in flexible multibody simulations. The use of this approach, however, has been limited to
small deformation problems. In this investigation, the computer implementation of the new
absolute nodal co-ordinate formulation and its use in the small and large deformation
analysis of flexible multibody systems that consist of interconnected bodies are discussed.
While in the floating frame of reference formulation a mixed set of absolute reference and
local elastic co-ordinates are used, in the absolute nodal co-ordinate formulation only
absolute co-ordinates are used. In the absolute nodal co-ordinate formulation, new
interpretation of the nodal co-ordinates of the finite elements is used. No infinitesimal or
finite rotations are used as nodal co-ordinates for beams and plates, instead, global slopes
are used to define the element nodal co-ordinates. Using this interpretation of the element
co-ordinates, beams and plates can be considered as isoparametric elements, and as a result,
exact modelling of the rigid body dynamics can be obtained using the element shape
function and the absolute nodal co-ordinates. Unlike the floating frame of reference
approach, no co-ordinate transformation is required in order to determine the element
inertia. The mass matrix of the finite elements is a constant matrix, and therefore, the
centrifugal and Coriolis forces are equal to zero when the absolute nodal co-ordinate
formulation is used. Another advantage of using the absolute nodal co-ordinate
formulation in the dynamic simulation of multibody systems is its simplicity in imposing
some of the joint constraints and also its simplicity in formulating the generalized forces
due to spring-damper elements. The results obtained in this investigation show an excellent
agreement with the results obtained using the floating frame of reference formulation when
large rotation–small deformation problems are considered.
7 1998 Academic Press

1. INTRODUCTION
The formulation of the equations of motion of flexible multibody systems using the finite
element method has been a challenging problem, particularly when conventional
non-isoparametric elements such as beams and plates are used. The nodal co-ordinates of
these widely used elements include infinitesimal rotations. As a result, exact modelling of
the rigid body dynamics cannot be obtained when these non-isoparametric elements are
used [1]. Such a limitation poses a serious problem when flexible multibody systems are
considered. Generally these systems consist of interconnected rigid and deformable bodies,
each of which may undergo large rotations. For this reason, several formulations that lead

0022–460X/98/300833 + 19 $30.00/0 7 1998 Academic Press

834 . .    .
to exact modelling of the rigid body inertia were proposed for the non-linear dynamic
analysis of flexible multibody systems. Among these formulations is the floating frame of
reference approach [2–6] which can be used to obtain accurate modelling of the rigid body
dynamics and it also leads to zero strain under an arbitrary rigid body motion of the
non-isoparametric finite elements. The floating frame of reference approach uses two sets
of co-ordinates to describe the dynamics of deformable bodies that undergo large reference
displacements. The large reference translations and rotations are described by a mixed set
of absolute Cartesian and orientation co-ordinates defined in a global inertial frame of
reference. The elastic displacements of the bodies are defined with respect to its co-ordinate
system using the nodal co-ordinates of the finite elements. The body frame of reference
is defined using an appropriate set of reference conditions that define a unique
displacement field [6]. The equations of motion obtained using the floating frame of
reference formulation exhibit a strong non-linear inertia coupling between the reference
and elastic co-ordinates. The mass matrix is highly non-linear and the inertia forces include
Coriolis and centrifugal forces which are quadratic in the velocities. The stiffness matrix,
on the other hand, takes a simple form and it is the same as the stiffness matrix that appears
in structural mechanics.
The use of two different types of frames of reference; global and local (inertial and
non-inertial), to describe two different sets of co-ordinates (reference co-ordinates and
elastic co-ordinates), leads to the complexity of the resulting inertia forces. If isoparametric
finite elements, which have absolute nodal co-ordinates defined in the inertial frame of
reference, are used to model the flexible bodies, much simpler expressions for the inertia
forces can be obtained. Furthermore, the shape function and the nodal co-ordinates of the
element can be used to obtain exact modelling of the rigid body dynamics provided that
the finite element shape functions have a complete set of rigid body modes. In the absolute
nodal co-ordinate formulation [7, 8], a new interpretation of the nodal co-ordinates is used
in order to develop new isoparametric beam and plate elements. Unlike the work of Simo
and Vu-Quoc [9], no finite rotations are used as nodal co-ordinates, and instead, global
slopes are used as nodal co-ordinates. The use of finite rotations as nodal co-ordinates can
lead to redundancy in representing the large rotation of the cross-section of the finite
element [6, 7].
In addition to the fact that the absolute nodal co-ordinate formulation automatically
captures the non-linear effects arising from the coupling between different modes of
displacements, the formulation of the joint constraints and forces becomes simpler when
this new approach is used in flexible multibody dynamics. It is the objective of this
investigation to examine and demonstrate the use of this new finite element procedure in
the small and large deformation analysis of flexible multibody systems that consist of
interconnected bodies. Comparison will be made with the floating frame of reference
formulation which is currently the most widely used computer procedure for the analysis
of flexible multibody systems. Throughout the analysis presented in this paper, a
two-dimensional beam element is used for demonstration purposes.
This paper is organized as follows. In section 2, the absolute nodal co-ordinate
formulation is reviewed, and the constant element mass matrix and non-linear stiffness
matrix are identified. In section 3, the formulation of the generalized forces, when the
absolute nodal co-ordinate formulation is used, is presented. In this section, some of the
fundamental differences between the absolute nodal co-ordinate formulation and other
existing finite element procedures are shown. Because of the use of global slopes as element
nodal co-ordinates, a new set of generalized moments must be used. In section 4, the
formulation of the joint constraints in the absolute nodal co-ordinate formulation is
discussed. Section 5, the relationship between co-ordinates used in the absolute nodal
   835
co-ordinate formulation and the floating frame of reference approach is presented.
Examples are presented in section 6 and the numerical results obtained using the absolute
nodal co-ordinate formulation are compared with the results obtained using the floating
frame of reference approach. Summary and conclusions drawn from the analysis developed
in this paper are presented in section 7.

2. ABSOLUTE NODAL CO-ORDINATE FORMULATION

In the mixed finite element formulations, displacements and displacement gradients are
used as nodal co-ordinates. These conventional finite element mixed formulations,
however, have serious limitations when flexible multibody applications are considered. For
instance, most of the mixed formulations were used in the framework of incremental
procedure and the shape functions employed often do not have a complete set of rigid body
modes. Furthermore, in structural dynamics applications mixed formulations are often
used with lumped masses. When a lumped mass formulation is used with conventional
beam elements, exact modelling of rigid body dynamics cannot be obtained [6]. In the
absolute nodal co-ordinate formulation used in this investigation, it is required that the
element shape function has a complete set of rigid body modes that can describe arbitrary
rigid body translational and rotational displacements. Global displacements and slopes are
used as nodal co-ordinates. By so doing, exact modelling of the rigid body dynamics can
be obtained provided that a consistent mass formulation is used.
The absolute nodal co-ordinate formulation is presented in several previous publications
[1, 6, 7, 8, 10, 11]. In this section, for the sake of completeness of the presentation, this
formulation is briefly reviewed. In the absolute nodal co-ordinate formulation, the
co-ordinates of the material points are defined in the global system. These absolute
co-ordinates, as shown in Figure 1, are defined in terms of the element shape function and
the vector of nodal co-ordinates as

$%
rx
r= = Se, (1)
ry

where r is the global position vector of an arbitrary point on the element, S is a global
shape function that includes a complete set of rigid body modes, and e is the vector of

n
Y
y
j
u
t
x
i A
e6
O
r
e2
R

e1 e5
X
Figure 1. Planar beam element.
836 . .   .
nodal co-ordinates that includes global displacements and slopes defined at the nodal
points of the element.

2.1.      

In this paper, a planar beam element is used as an example to demonstrate the use of
the finite element absolute nodal co-ordinate formulation in flexible multibody
applications. Since the co-ordinates of the material points in this formulation are defined
in a global frame of reference, there is no reason to use different polynomials to interpolate
the displacement components. In this investigation, a cubic polynomial is used for both
components of the displacement. In this case, the element shape function and the vector
of nodal co-ordinates are defined as [6, 8, 11]

$
1 − 3j 2 + 2j 3 0 l(j − 2j 2 + j 3)
S=
0 1 − 3j 2 + 2j 3 0

%
0 3j 2 − 2j 3 0 l(j 3 − j 2) 0
2 3 , (2)
l(j − 2j + j ) 0 3j 2 − 2j 3 0 l(j 3 − j 2)

e = [e1 e2 e3 e4 e5 e6 e7 e8 ]T, (3)

where the elements of the vector of nodal co-ordinates are defined as

1rx (x = 0) 1ry (x = 0)
e1 = rx (x = 0), e2 = ry (x = 0), e3 = , e4 = ,
1x 1x

1rx (x = l) 1ry (x = l) (4)

e5 = rx (x = l), e6 = ry (x = l), e7 = , e8 = ,
1x 1x

where x is the spatial co-ordinate along the element axis. Note that in the absolute nodal
co-ordinate formulation no infinitesimal rotations are used as nodal co-ordinates, instead,
global slopes are used. The initial values of the global slopes in the undeformed reference
configuration can be determined using simple rigid body kinematics by utilizing the fact
that equation (1) can be used to obtain exact modelling of the kinematics of rigid bodies.
For instance, in an arbitrary undeformed reference configuration defined by the
translations rx (x = 0) and ry (x = 0) and the rigid body rotation u, the global position of
an arbitrary point on the beam can be written as

$ % $ %
rx (x) r (x = 0) + x cos u
r(x) = = Se = x . (5)
ry (x) ry (x = 0) + x sin u

It follows that the global slopes in the undeformed reference configuration are defined
as [6, 10, 11]

e3 = e7 = cos u, e4 = e8 = sin u. (6)

A similar procedure can be used to determine the global slopes in the case of
three-dimensional elements.
   837
2.2.  
The kinetic energy of the beam element is defined as

g 0g 1
1
T= rṙTṙ dV = 12 ėT rSTS dV ė = 12 ėTMa ė, (7)
2 V V

where V is the volume, r is the mass density of the beam material, and Ma is the mass
matrix of the element. Note that the mass matrix in equation (7) is symmetric and constant,
and it is the same matrix that appears in linear structural dynamics. Using the shape
function of equation (2), the mass matrix of the element can be evaluated as [6, 8, 11]

K 13 11l 9 13l L
0 0 0 − 0
G 35 210 70 420 G
G 13 11l 9 13l G
G 0 0 0 −
35 210 70 420 G
G G
l2 13l l2
G 0 0 − 0 G
105 420 140
G G
2
G l 13l l2 G
0 0 −
G 105 420 140 G
G G
V g
Ma = rSTS dV = m
G
G
13
35
0 −
11l
210
0 G,
G
G 13 11l G
G symmetric 0 −
35 210 G
G G
l2
G 0 G
105
G G
G l2 G
k 105 l

(8)

where m is the mass of the beam and l is its length. It can be demonstrated that the use
of this mass matrix leads to exact modelling of the rigid body inertia [8].

2.3.  

While the absolute nodal co-ordinate formulation leads to a simple expression for the
inertia forces, the use of this formulation results in a relatively complex expression for the
elastic forces. In order to demonstrate this fact, a simple linear elastic model based on the
classical beam theory is used in this section. If point O shown in Figure 1 is used as the
reference point, the displacements of an arbitrary point on the beam relative to point O
may be written as

$% $ %
ux (S1 − S1O )e
u= = , (9)
uy (S2 − S2O )e
838 . .   .
where S1 and S2 are the rows of the element shape function matrix, and S1O and S2O are
the rows of the element shape function matrix defined at point O. In order to define these
relative displacements in the element co-ordinate system, two unit vectors i and j along
the element axes are defined as

$% $%
ix r − rO jx
i= = A , j= = k × i, (10)
iy =rA − rO = jy

where k is a unit vector along the Z-axis. The longitudinal and transverse deformations
of the beam can then be defined as [6, 8, 11]

$% $ % $ %
ul uTi − x u i + uy iy − x
ud = = = xx . (11)
ut uTj ux jx + uy jy

The strain energy of the beam element due to the longitudinal and transverse
displacements is given by

g0 0 1 0 11
l 2 2
1 1ul 1 2ut
U= Ea + EI dx = 12 eTKa e, (12)
2 0
1x 1x 2

where E is the modulus of elasticity, a is the cross-sectional area, I is the second moment
of area of the beam element, and Ka is the element stiffness matrix. This matrix is a
non-linear function of the nodal co-ordinates. It can be shown that the strain energy can
be expressed in terms of the following stiffness shape integrals [6, 8, 11]:

g $0 1 0 1% g $0 1 0 1%
1 T 1 T
Ea 1S1 1S1 Ea 1S1 1S2
A11 = dj, A12 = dj,
l 0
1j 1j l 0
1j 1j

g $0 1 0 1% g $0 1 0 1%
1 T 1 T
Ea 1S2 1S1 Ea 1S2 1S2
A21 = dj, A22 = dj,
l 0
1j 1j l 0
1j 1j

g $0 1 0 1% g $0 1 0 1%
1 T 1 T
EI 1 2S1 1 2S 1 EI 1 2S 1 1 2S 2
B11 = dj, B12 = dj, (13)
l3 0
1j 2 1j 2 l3 0
1j 2 1j 2

g $0 1 0 1% g $0 1 0 1%
1 T 1 T
EI 1 2S 2 1 2S 1 EI 1 2S 2 1 2S 2
B21 = dj, B22 = dj,
l3 0
1j 2 1j 2 l3 0
1j 2 1j 2

g0 1 g0 1
1 T 1 T
1S1 1S2
A1 = Ea dj, A2 = Ea dj,
0
1j 0
1j
   839
where the explicit forms of these matrices obtained using the shape function of equation
(2) are given in the Appendix. Using these stiffness shape integrals, the generalized elastic
forces of the element can be calculated from [6, 8, 11]

0 1
T
1U
= A11 eix2 + A22 eiy2 + (A12 + A21 )eix iy − A1 ix − A2 iy + B11 ejx2 + B22 ejy2
1e

0 1
T
1ix
+ (B12 + B21 )ejx jy + (eTA11 eix + 12 eT(A12 + A21 )eiy − AT1 e)
1e

01
T
1iy
+ (eTA22 eiy + 12 eT(A12 + A21 )eix − AT2 e)
1e

0 1
T
1jx
+ (eTB11 ejx + 12 eT(B12 + B21 )ejy )
1e

01
T
1jy
+ (eTB22 ejy + 12 eT(B12 + B21 )ejx ) , (14)
1e

where

0 1 01
T T
1ix 1iy
=
1e 1e

= D[ − (e6 − e2 )2 (e5 − e1 )(e6 − e2 ) 0 0 (e6 − e2 )2 −(e5 − e1 )(e6 − e2 ) 0 0]T,

01 0 1
T T
1iy 1jx
=−
1e 1e

= D[(e5 − e1 )(e6 − e2 ) −(e5 − e1 )2 0 0 −(e5 − e1 )(e6 − e2 ) (e5 − e1 )2 0 0]T,

(15)
1
D= .
((e5 − e1 )2 + (e6 − e2 )2)3/2

2.4.   

Using the principle of virtual work in dynamics and the expression of the kinetic and
strain energies given by equations (7) and (12), the equation of motion of the finite element
can be written as
Ma ë = Q, (16)
where Q is the vector of generalized external nodal forces including the elastic forces. Note
that centrifugal and Coriolis forces are equal to zero since the mass matrix is constant.
The equations of motion of the deformable body can be obtained by assembling the
equations of its elements using a standard finite element procedure.

3. FORMULATION OF THE GENERALIZED EXTERNAL FORCES

It is clear from the analysis presented in the preceding section that there are several
fundamental differences between the absolute nodal co-ordinate formulation and some of
840 . .   .
the existing finite element procedures. One of these differences is the fact that there is no
need to use co-ordinate transformation in order to determine the element mass matrix.
Another difference is attributed to the formulation of the stiffness matrix which is highly
non-linear in the case of the absolute nodal co-ordinate formulation even in the case of
simple linear elastic model.
Another fundamental difference is due to the nature of the co-ordinates used in the
absolute nodal co-ordinate formulation. These co-ordinates do not include infinitesimal
or finite rotations. As such, attention must be paid to the definition of the generalized
forces associated with the global slopes of the finite element. In this section, the definition
of the generalized forces in the absolute nodal co-ordinate formulation is discussed.

3.1.  

The virtual work due to an externally applied force F acting on an arbitrary point on
the element is given by FTdr, where r is the position vector of the point of application of
the force and dr is the virtual change in the vector r. In order to obtain the generalized
forces associated with the absolute nodal co-ordinates it is necessary to express dr in terms
of the virtual displacements of these nodal co-ordinates. To this end, one can write

FTdr = FTSde = QTFde, (17)

where QF = STF is the vector of generalized forces associated with the element nodal
co-ordinates. For example, the virtual work due to the distributed gravity of the finite
element can be obtained using the shape function of equation (2) as

g $ %
1 l 1 l
[0 −rg]Sde dV = mg 0 − 0 − 0 − 0 de, (18)
v
2 12 2 12

which defines the vector of generalized distributed gravity forces as

$ %
T
1 l 1 l
QF = mg 0 − 0 − 0 − 0 . (19)
2 12 2 12

3.2. 
When a moment M acts at a cross-section of the beam, the virtual work due to this
moment is given by Mda, where a is the angle of rotation of the cross-section. The
orientation of a co-ordinate systen whose origin is rigidly attached to this cross-section
(see Figure 1) can be defined using the following transformation matrix:

K
G1rx −
1ryL
G
$ % 0 1 0 1
2 2
cos a −sin a 1 1x 1x 1rx 1ry
= 1/2 G G, d= + . (20)
sin a cos a d 1ry 1rx 1x 1x
G
k1x 1x l
G
Using the elements of the planar transformation matrix given in the preceding equation,
one has

0 1 0 1
1ry 1rx
sin a = d−1/2 , cos a = d−1/2 . (21)
1x 1x
   841

Y
Element i
Element j

rP

X
Figure 2. Revolute (pin) joint between two elements.

Using these two equations, it can be shown that

0 1 0 1
1rx 1ry 1r 1r
d − yd x
1x 1x 1x 1x
da = . (22)
d
If the concentrated moment M is applied, for example, at node O of the element, the
generalized forces due to this moment are defined as

$ %
T
−Me4 Me3
QM = 0 0 0 0 0 0 . (23)
d d

3.3. - 

The formulation of the generalized forces due to a springer-damper element connecting
two finite elements is very simple as compared to the floating frame of reference
formulation which leads to a highly non-linear complex expression for these forces [5]. In
the absolute nodal co-ordinate formulation, the generalized forces due to a spring-damper
element take a simple form due to the fact that absolute co-ordinates are used. If a and
b are the nodes to which the ends of the spring-damper element are attached, the
generalized forces acting at node b simply take the form

$ % $ %
e1a − e1b ė1a − ė1b
QSD = k a b +c , (24)
e2 − e2 ė2a − ė2b

where k and c are the spring and damping coefficients, respectively.

4. FORMULATION OF CONSTRAINTS
The formulation of many of the constraint equations that describe mechanical joints in
flexible multibody systems becomes relatively simple when the absolute nodal co-ordinate
formulation is used. In many cases, these constraint equations take a complex non-linear
form when the floating frame of reference approach is used. This is mainly due to the fact
that in the floating frame of reference formulation, two sets of co-ordinates (reference and
842 . .   .
elastic) defined in two different frames of reference (global and body) are used. In the
absolute nodal co-ordinate formulation, only one set of absolute co-ordinates defined in
one global co-ordinate system is used. As a consequence, many of the constraint equations
become simple and linear. For instance, the revolute joint constraints which are highly
non-linear in the floating frame of reference formulation [5, 6] become simple and linear
when the absolute nodal co-ordinate formulation is used. Figure 2 shows two elements i
and j which are connected by a revolute joint at point P. The constraint equations for the
revolute joint can be written as
riP = rjP , (25)
which can be written in terms of the element co-ordinates as
SiP ei = SjP ej, (26)
where S and S are the shape functions of the elements i and j evaluated at point P, and
i
P
j
P
ei and e are the vectors of nodal co-ordinates of the two elements. If point P is selected
j

as a nodal point on the two elements, the constraint equation of the revolute joint reduces
to

$ %
e5i − e1j
= 0, (27)
e6i − e2j

where e5i and e6i are the absolute translational nodal co-ordinates of element i at node P,
and e1j and e2j are the absolute translational nodal co-ordinates of element j at node P.

5. COMPARISON WITH THE FLOATING FRAME OF REFERENCE FORMULATION

In the floating frame of reference formulation, not all co-ordinates represent absolute
variables, since the configuration of the body is described using a mixed set of absolute
reference and local deformation co-ordinates. The reference co-ordinates define the
location and the orientation of a selected body co-ordinate system. The deformation of
the body is described using a set of local shape functions and a set of deformation
co-ordinates defined in the body co-ordinate system. In the floating frame of reference
formulation, it is assumed that there is no rigid body motion between the body and its
co-ordinate system. Using Figure 1 and the reference and deformation co-ordinates, the
global position vector of an arbitrary point on the centreline of the beam element can be
written as [6]
r = R + Au, (28)
where R = R(t) defines the global position vector of the origin of the selected beam
co-ordinate system, A = A(t) is the transformation matrix that defines the orientation of
the selected beam co-ordinate system with respect to the inertial frame, and u = u(x, t) is
the local position vector of the arbitrary point defined with respect to the origin of the
beam co-ordinate system. The local position vector u may be represented in terms of local
shape functions Sl (x) as
u(x, t) = Sl (x)qf (t), (29)
where qf (t) is the vector of time dependent deformation co-ordinates which can also be
used in the finite element formulation to interpolate the local position as well as the
deformation. When the kinematic description of equation (28) is used, it is assumed that
there is no rigid body motion between the beam and its co-ordinate system. As a
   843
consequence, it is required that the local shape function matrix Sl (x) contains no rigid body
modes. Using equations (28) and (29), the motion of the flexible beam can be described
using the floating frame of reference formulation as
r = R + ASl qf , (30)
where the vector qf (t) describes the local position and the deformation of an arbitrary point
[6, 10], and the vector

$ %
R(t)
qr (t) = (31)
u(t)

describes the reference motion. In equation (31), u is the angle that defines the orientation
of the beam co-ordinate system. Therefore, the vector of generalized co-ordinates of the
beam used in the floating frame of reference formulation can be written in a partitioned
form as
q = [RT u qTf ]T = [qTr qTf ]T. (32)
Using equation (30) and the co-ordinate partitioning of equation (32), it can be shown
that the mass matrix of the deformable beam in the case of the floating frame of reference
formulation can be written in a partitioned form as [6]

$ %
mrr mrf
Mf = . (33)
mfr mff

Unlike the absolute nodal co-ordinate formulation which leads to a simple and constant
mass matrix, the mass matrix in the preceding equation is highly non-linear in the
co-ordinates q = [qTr qTf ]T as a result of the dynamic coupling between the reference
co-ordinates qr and the deformation co-ordinates qf . In the case of planar motion, one has

$ %
cos u −sin u
qr = [Rx Ry u]T, A= . (34)
sin u cos u

In this case of planar motion, it can be shown that the non-linear mass matrix and the
Coriolis and centrifugal forces of the finite element can be expressed in terms of the
following constant inertia shape integrals [6]:

S
=
gV
rSl dV, mff =
gV
rSTl Sl dV, S=
g
V
rSTl ISl dV, (35)

where r and V are the mass density and volume of the element, and

$ %
0 1
I= . (36)
−1 0

By establishing the relationship between the co-ordinates used in the floating frame of
reference formulation and the co-ordinates used in the absolute nodal co-ordinate
formulation, the non-linear mass matrix of equation (33) can be obtained using the
constant mass matrix of equation (8) [10].
844 . .   .
5.1.  
Using equation (28), the global position vector of an arbitrary point on the beam element
can be written using the floating frame of reference formulation as

$% $ %
rx R + ux cos u − uy sin u
r(x, t) = = x , (37)
ry Ry + ux sin u + uy cos u

where ux and uy are the position co-ordinates of the arbitrary point defined with respect to
the beam co-ordinate system. It follows in the case of a slender beam element that

1rx 1ux 1u 1ry 1ux 1u

= cos u − y sin u, = sin u + y cos u. (38)
1x 1x 1x 1x 1x 1x

The slope relationship plays a fundamental role in defining the relationship between the
co-ordinates used in the absolute nodal co-ordinate formulation and the co-ordinates used
in the floating frame of reference formulation.

5.2. - 

In the remainder of this section, the relationship between the co-ordinates used in the
floating frame of reference formulation and the co-ordinates used in the absolute nodal
co-ordinate formulation is presented [10]. In the case of the absolute nodal co-ordinate
formulation, the global element shape function defined by equation (2) is used. In the
floating frame of reference formulation, it is assumed that the origin of the beam
co-ordinate system is located at point O and one of the axes connects points O and A.
In this case, the local shape function can be obtained from the global shape function of
equation (2) as

$ %
l(j − 2j 2 + j 3) 0 3j 2 − 2j 3 l(j 3 − j 2) 0
Sl = 2 3 . (39)
0 l(j − 2j + j ) 0 0 l(j 3 − j 2)

Note that this local shape function does not include any rigid body modes. The vector
qf in this case can be defined as

qf = [q1 q2 q3 q4 q5 ]T, (40)

where q3 is the local x co-ordinate of the node at A defined in the beam co-ordinate system,
and

1ux (x = 0) 1uy (x = 0) 1ux (x = l) 1uy (x = l)

q1 = , q2 = , q4 = , q5 = . (41)
1x 1x 1x 1x

The vector e of equation (3) used in the absolute nodal co-ordinate formulation can be
expressed in this case in terms of the components of the vector

q = [Rx Ry u q1 q2 q3 q4 q5 ]T (42)
   845
of the floating frame of reference formulation using equation (38) as

K L K
G e1 G G Rx L
G
G e2 G G Ry G
G e3 G G q1 cos u − q2 sin uG
G e4 G G q1 sin u + q2 cos uG
e = G G=G G. (43)
G e5 G G Rx + q3 cos u G
G e6 G G Ry + q3 sin u G
G e7 G G q4 cos u − q5 sin uG
G G G G
k e8 l k q4 sin u + q5 cos ul

Using this vector, it can be shown that

Se = R + ASl qf = r. (44)

This equation demonstrates the equivalence of the kinematic descriptions used in the
floating frame of reference formulation and the absolute nodal co-ordinate formulation.
Therefore, the co-ordinate transformation of equation (43) can be used to obtain the
non-linear mass matrix and the inertia shape integrals used in the floating frame of
reference formulation from the constant mass matrix used in the absolute nodal
co-ordinate formulation, as demonstrated in reference [10].

6. APPLICATIONS
In order to demonstrate the use of the absolute nodal co-ordinate formulation in the
dynamic simulation of flexible multibody systems, two examples are considered in this
section. The results obtained using the absolute nodal co-ordinate formulation are
compared with the results obtained using the floating frame of reference formulation. The
two examples considered are the free falling of a flexible pendulum under its own weight,
and a flexible slider–crank mechanism driven by a moment applied to the crankshaft. Both
the crankshaft and the connecting rod of the slider–crank mechanism are assumed to be
flexible bodies. It is important, however, to point out that the floating frame of reference
formulation can only be used in the case of small deformation because the deformation
of the bodies is expressed in terms of infinitesimal rotations and linear mode shapes. The
absolute nodal co-ordinate formulation, on the other hand, can be used in the small as
well as in the large deformation analysis.

gA

X
Figure 3. Free falling of a flexible pendulum.
846 . .   .
0

–1

Angle (rad)
–2

–3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time (s)
Figure 4. Angular orientation of the pendulum: ——, absolute nodal co-ordinate formulation; – – –, floating
frame of reference formulation.

6.1.  

The first example considered in this section is the free falling of the flexible pendulum
shown in Figure 3. The pendulum, which is horizontal in its initial position, falls under the
effect of gravity. The beam has a length of 0·4 m, a cross-sectional area of 0·0018 m2, a second

0.005
Transverse displacement (m)

0.004
0.003
0.002
0.001
0.000
–0.001
–0.002
–0.003
–0.004
–0.005
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time (s)
Figure 5. Transverse deformation of the tip point of the pendulum: key as in Figure 4.

Y
M
B
O
X

Figure 6. Slider–crank mechanism.

   847
4 3
moment of area of 1·215 E-08 m , a mass density of 5540 kg/m and a modulus of elasticity
of 1·0 E 09 N/m2. The beam is divided into 10 elements. In the floating frame of reference
formulation, 10 elastic modes are used to describe flexibility of the pendulum rod. The
body frame of reference of the flexible pendulum is assumed to be rigidly attached to its
end at the pin joint. Note that in the absolute nodal co-ordinate formulation 42 degrees
of freedom are used, as compared to 13 co-ordinates in the floating frame of reference
formulation; 10 of them describe the elastic deformation.
Figure 4 shows the angular orientation of the flexible pendulum versus time obtained
using the two formulations. A very good agreement can be observed between the two
methods. Figure 5 shows the transverse displacement of the tip node of the pendulum
versus time. The results presented in this figure show a good agreement between the
absolute nodal co-ordinate formulation and the floating frame of reference formulation
in the case of small deformation analysis.

6.2.  – 

The second example used in this section to demonstrate the use of the absolute nodal
co-ordinate formulation in the simulation of flexible multibody systems is the flexible
slider–crank mechanism shown in Figure 6. The connecting rod is assumed to be much
more flexible than the crankshaft and the slider block is assumed to be rigid and massless.
In the initial position, both the connecting rod and crankshaft are assumed to be
horizontal. The mechanism is assumed to be driven by a moment applied at the crankshaft.
The crankshaft has a length of 0·152 m, a cross-sectional area of 7·854E-05 m2, a second
moment of area of 4·909E-10 m4, a mass density of 2770 kg/m3 and a modulus of elasticity
of 1·0 E 09 N/m2. The connecting rod is a beam of length 0·304 m, and has the same
cross-sectional dimension and material properties as the crankshaft, with the exception of
the modulus of elasticity which is assumed to be 0·5 E 08 N/m2. In the dynamic model
used in this study, the crankshaft is divided into three finite elements and the connecting
rod is divided into eight elements. In the floating frame of reference formulation, three
mode shapes are used to describe the flexibility of the crankshaft and five mode shapes
are used for the connecting rod.
Two simulation cases were performed. In the first case, the moment applied at the
crankshaft is given by
M(t) = [0·01(1 − e−t/0·167)] Nm. (45)

0.5
Slider position (m)

0.4

0.3

0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Time (s)
Figure 7. Co-ordinate of the slider block (moment defined by equation (45)): key as in Figure 4.
848 . .   .
0.5

0.4

Slider position (m)

0.3

0.2

0.1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Time (s)
Figure 8. Co-ordinate of the slider block (moment defined by equation (46)): key as in Figure 4.

In the second case, the moment is assumed to be

6
[0·01(1 − e−t/0·167)] Nm, t E 0·7,
M(t) = (46)
0, t q 0·7.

Two variables are used to compare the results obtained using the absolute nodal
co-ordinate formulation and the floating frame of reference formulation. These are the X
position of the slider block and the transverse deformation of the mid-point of the
connecting rod. Figures 7 and 8 show the slider block position in the two cases of the
applied moments. These two figures show good agreement between the results obtained
using the absolute nodal co-ordinate formulation and the floating frame of reference
approach. Figures 9 and 10 show the transverse deformation of the mid-point of the
connecting rod. In the first case of the applied moment, when the velocity of the system
increases as well as the inertia forces, the deformation becomes relatively large, and

0.02
Connecting rod deflection (m)

0.01

0.00

–0.01

–0.02

–0.03

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Time (s)
Figure 9. Deformation of the mid-point of the connecting rod (moment defined by equation (45)): key as in
Figure 4.
   849
0.0015

Connecting rod deflection (m)

0.0010

0.0005

0.0000

–0.0005

–0.0010

–0.0015
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Time (s)
Figure 10. Deformation of the mid-point of the connecting rod (moment defined by equation (46)): key as
in Figure 4.

differences between the solutions obtained using the two formulations can be observed.
In the second case of the applied moment the transverse deformation remains relatively
small. In this case, excellent agreement between the two formulations can be observed, as
shown in Figure 10.

7. SUMMARY AND CONCLUSIONS

In the absolute nodal co-ordinate formulation, a new interpretation for the nodal
co-ordinates is used. By using this new interpretation of the co-ordinates, a constant mass
matrix can be obtained and as a result the Coriolis and centrifugal forces are equal to zero.
The elastic forces, on the other hand, are highly non-linear functions of the element
co-ordinates. The absolute nodal co-ordinate formulation can be effectively used in the
large deformation problems [11] as well as flexible multibody applications as demonstrated
in this paper. In addition to the constant simple mass matrix that appears in this
formulation, the formulation of some of the joint constraints as well as forces can be very
simple as compared to the floating frame of reference approach. Because of the nature of
the co-ordinates used in the floating frame of reference formulation, such a method has
only been used in the small deformation analysis of flexible multibody systems. The
absolute nodal co-ordinate formulation does not suffer from this limitation and can be
used in the small and large deformation analysis of flexible multibody systems. The
applications used in this paper to compare the results obtained using the absolute nodal
co-ordinate formulation and the results obtained using the floating frame of reference
approach show excellent agreement between the two methods in the analysis of small
deformations. Discrepancies can be observed between the results obtained using the two
methods as the deformation increases.

ACKNOWLEDGMENT
This research was supported by the U.S. Army Research Office, Research Triangle Park,
NC.
850 . .   .
REFERENCES
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incremental approach and exact rigid body inertia.
2. H. B and F. P 1992 Elastische Mehrkörpersysteme. Stuttgart: Teubner Publisher.
3. R. K. C and A. R. D 1977 AIAA Journal 15, 1684–1690. Hamilton’s principle: finite
element methods and flexible body dynamics.
4. B. F. D V 1976 International Journal of Engineering Science 14, 895–913. The dynamics
of flexible bodies.
5. W. K, D. S and R. S 1995 Paper IAF-95-Ak.04, 46th International
Astro. Conference, Oslo, Norway. Analysis and design of flexible and controlled multibody with
SIMPACK.
6. A. A. S 1998 Dynamics of Multibody Systems. New York: Cambridge University Press,
second edition.
7. A. A. S 1996 Technical Report no. MBS96-1-UIC, Department of Mechanical
Engineering, University of Illinois at Chicago. An absolute modal coordinate formulation for the
large rotation and deformation analysis of flexible bodies.
8. A. A. S, H. A. H and J. L. E 1997 Proceedings of the 16th ASME Biennial
Conference on Mechanical Vibration and Noise, Sacramento, CA. Absolute nodal coordinate
formulation.
9. J. C. S and L. V-Q 1986 ASME Journal of Applied Mechanics 53, 849–863. On the
dynamics of flexible beams under large overall motions–the plane case: parts I & II.
10. A. A. S and R. S 1998 International Journal of Non-Linear Mechanics 33,
417–432. Equivalence of the floating frame of reference approach and finite element
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11. A. A. S, H. A. H and J. L. E, (in press), ASME Journal of Mechanical
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APPENDIX: STIFFNESS SHAPE INTEGRALS

The definition of the matrices that appear in the elastic forces can be made simpler if
the nodal co-ordinates are rearranged as

e = [e1 e3 e5 e7 e2 e4 e6 e8 ]T = [ex ey ]T. (A1)

Define the matrix A and B as

K
G 6 1 6 1 L G
K 12
G 3 6 12 6 L G
− 2 − 3
G 5l 10 5l 10 G G l l l l2 G
G 1 2l 1 l G G 6 4 6 2 G
G 10 − − G G l2 −2
15 10 30 l l l G
A = Ea G G, B = EI G G.
G− 6 − 1 6
−
1 G G−123 − 62 123 − 62 G
G 5l 10 5l 10 G G l l l l G
G 1 l 1 2l G G 6 2 6 4 G
G − − −2
k 10 30 10 15 Gl
G
k l
2
l l l Gl

(A2)
These matrices can be considered as the axial and bending stiffness matrices that appear
in linear structural dynamics. By using the arrangement defined in equation (A1) and the
   851
matrices in equation (A2), the stiffness shape integrals that appear in the expression of the
elastic forces are

$ % $ % $ % $ %
A 0 0 0 0 A 0 0
A11 = , A22 = , A12 = , A21 = ,
0 0 0 A 0 0 A 0

$ % $ % $ % $ %
B 0 0 0 0 B 0 0 (A3)
B11 = , B22 = , B12 = , B21 =
0 0 0 B 0 0 B 0

and
A1 = [−Ea Ea 0 0 0 0 0 0]T, A2 = [0 0 0 0 −Ea Ea 0 0]T.
(A4)