Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

2, 40, 2002

INTERACTIONS BETWEEN VIBRATIONS OF FLEXIBLE

LINKS AND BASE MOTION OF MANIPULATORS

Iwona Adamiec-Wójcik

Department of Mechanics and Computational Methods

Technical University of Łódź in Bielsko-Biała

e-mail: iadamiec@pb.bielsko.pl

Thepaperpresents anapplicationof theRigidFiniteElementMethod to
modelling of flexible links of amanipulator. The equations ofmotion are
derived assuming that the basemotion is not knownand thus it depends
on generalised co-ordinates. Numerical calculations are carried out in
order to show not only the influence of the basemotion on vibrations of
the flexible links but also the reverse connection.

Key words: dynamic analysis, flexibility of links, changing configuration

1. Introduction

Dynamics of flexible systems has been widely investigated over decades.
The possibility of changing configuration is the feature of flexible systems
causingmost problems. Large basemotion causes additional centrifugal forces
and vibrations of the flexible links. Most popular approaches to solve the
problem are based on the Finite Element Method (FEM) (Gao et al., 1989;
Bircout et al., 1990; Du et al., 1996). In order to reduce the number of degrees
of freedom and thus the dimension of the non-linear differential equations the
modal method is used by Kane et al. (1987), Du et al. (1992).

A different approach used for discretisation of flexible links is based on
the Rigid Finite Element Method (RFEM) described by Kruszewski et al.
(1975). Themethod has been successfully applied to dynamic analysis of both
planar (Wojciech, 1984; Adamiec and Wojciech, 1993) and spatial systems
with changing configuration (Wojciech, 1990; Adamiec, 1992; Wittbrodt and
Wojciech, 1995). Not only the flexibility of links but also friction in joints
(Wojciech, 1995) as well as the flexibility of connecting elements (Wojciech,



450 I.Adamiec-Wójcik

1996) have been investigated. The results obtained using the RFEM have
been compared with those obtained using the FEM (Wojciech, 1996; Plosa
and Wojciech, 2000) as well as with experiments (Adamiec, 1992; Plosa and
Wojciech, 2000). This comparison shows that the RFEMgives reliable results
despite its simplicity and can be used to simulate the dynamics of systems
with changing configuration.

In the above papers the base motion is assumed to be known. However,
in such systems not only the base motion influences the vibrations of flexible
links but there is also an opposite effect – the base motion can be influenced
by the vibrations of the flexible links. In the present paper the RFEM is
applied in order to derive the equations of motion for flexible systems with
changing configurations assuming that the base motion is not known. In this
way it will be possible to investigate the influence of flexible vibrations on
the base motion. Since the link after discretisation is treated as a system
of rigid bodies connected by joints, the conventional kinematic description
widely employed for rigid manipulators can be used in a straightforward way
for flexible mechanisms (Wojciech, 1996). The RFEM enables us to obtain
the equations of motion of a system with changing configuration by merely
a slight modification of the equations of motion of a system with all rigid
links. Moreover, the equations of motion of a rigid system can be obtained
as a special case of generalised equations of motion. Thus, using the same
equations of motion, both vibrations of the flexible link caused by known
motion of the base and the influence of vibrations on the base motion in the
case of force input will be analysed in the paper.

2. Rigid finite element method and description of the model

The RFEM is based upon discretisation of continuous systems into un-
deformable bodies called rigid finite elements. These elements are connected
with spring-damping elements, the characteristics of which are linear. In order
to describe the motion of the system a local co-ordinate system is connected
with each rfe (rigid finite element). The beginning of the system is placed in
the mass centre and the axes coincide with the principal inertial axes before
deformation of the link.

The co-ordinate system {r} is connected with the base (Fig.1) and the
vibrations of the system of rfe’s caused by the base motion are taken into
consideration.



Interactions between vibrations of... 451

Fig. 1. Co-ordinate systems: {0} – inertial system, {r} – system connected with the
moving base, {i′} – local system connected with the rfe

i
before deformation of the

link

It is assumed that the base motion is not known and it depends on ge-

neralised co-ordinates defined by the vector qb = [q
(b)
1 , ...,q

(b)
n ]
⊤. Then the

basemotion can bedescribed using a transformationmatrix dependent on the
vector qb in the following form

B0(qb)=
0
rT(qb)=B

(b)
1 B

(b)
2 ...B

(b)
m (2.1)

The matrices B
(b)
i are the transformation matrices which are used in the de-

scription of rigid systems by Craig (1988).
The flexible system is divided into n+1 rfes, and the rfe0 is the last link

of the base (Fig.2).
The vector of generalised co-ordinates of each rfe takes the following form

q
(f)
i = [xi1,xi2,xi3,ϕi1,ϕi2,ϕi3]

⊤ (2.2)

where
xi1,xi2,xi3 – co-ordinates of the mass centre of the rfei
ϕi1,ϕi2,ϕi3 – generalised co-ordinates describing bending and

torsion of the rfei.

Thus, the generalised co-ordinates of the whole flexible system form the
vector

q
(f) = [q

(f)⊤
1 ,q

(f)⊤
2 , ...,q

(f)⊤
n ]

⊤ (2.3)

The position of any rfe before deformation can be defined if the matrices
of transformation from the system connected with rfei to the system of the



452 I.Adamiec-Wójcik

Fig. 2. System of rfes and sdes

rfe0
r
i′T = const, i = 1,2, ...,n are known. The matrix corresponding to the

transformation from the co-ordinate system after deformation to that before
the deformation can be written as follows

i′

iT=

















ci2ci3 −ci2si3 si2 xi1

si1si2ci3+ ci1si3 −si1si2si3+ ci1ci3 −si1ci2 xi2

−ci1si2ci3+si1si3 ci1si2si3+si1ci3 ci1ci2 xi3

0 0 0 1

















(2.4)

where: xi1,xi2,xi3,ϕi1,ϕi2,ϕi3 are the generalised co-ordinates of the rfei,
and sij =sinϕij, cij =cosϕij for i =1, ...,n; j =1,2,3.
Thematrix of transformation from the system connected with the rfei to

the inertial system can be determined from the formula

B
(f)
i =

0
rT
r
i′T
i′

iT=B0(qb)
r
i′T
i′

iT(q
(f)
i )=B0(qb)B

′(f)
i (q

(f)
i ) (2.5)

where B0 is thematrix of transformation from the rfe0 system to the inertial
system defined by the formula (2.1), and

B
′(f)
i (q

(f)
i )=

r
i′T
i′

iT(q
(f)
i )

3. Equations of motion

In order to derive the equations of motion the Lagrange equations of the
second order are used

d

dt

∂T

∂q̇l
−
∂T

∂ql
+
∂V

∂ql
= Ql l =1, ...,m+6n (3.1)



Interactions between vibrations of... 453

where
T,V – kinetic and potential energy of the system, respectively
Ql – generalised forces,
ql – generalised coordinate which is a component of the following

vector

qs = [q
(b)
1 , ...,q

(b)
m ,x11,x12,x13,ϕ11,ϕ12,ϕ13, ...,xn1,xn2,xn3,ϕn1,ϕn2,ϕn3]

⊤

(3.2)
The equations of motion require a definition of the kinetic and potential

energies of the system.
The kinetic energy includes the energies of both parts, the base and the

flexible part
T = Tb+Tf (3.3)

Having used the transformation matrices the following vectors describing the
position of any point of the system can be defined

0
ir= ri =B

(b)
1 B

(b)
2 ...B

(b)
i
i
rb for any point of the base

0
ir= ri =B0(qb)

r
i′T
i′

iT(q
(f)
i )

i
ri =B0B

′(f)
i
i
ri for any point of the rfei

(3.4)
Thus, the kinetic energy of the base is equal to

Tb =
m
∑

i=1

Tbi =
1

2

m
∑

i=1

tr
(

Ḃ
(b)
i HbiḂ

(b)⊤
i

)

(3.5)

where B
(b)
i and Hbi are the transformation and inertial matrices, respectively,

defined according to the rules for rigid systems.
The kinetic energy of the flexible part is a sum of the energies of all rfes,

and the energy of each rfe can be calculated as follows

Tfk =
1

2

∫

mk

ṙ
⊤

k ṙk dmk =
1

2

∫

mk

tr(ṙkṙ
⊤

k ) dmk =

=
1

2

∫

mk

tr(Ḃ
(f)
k
k
rk
k
r
⊤

k Ḃ
(f)⊤
k ) dmk = (3.6)

=
1

2
tr
(

Ḃ
(f)
k

∫

mk

k
rk
k
r
⊤

k dmk Ḃ
(f)⊤
k )=

1

2
tr(Ḃ

(f)
k HfkḂ

(f)⊤
k )

where

Hfk =

∫

mk

[kxk1,
kxk2,

kxk3,1]
⊤[kxk1,

kxk2,
kxk3,1] dmk



454 I.Adamiec-Wójcik

If the local system is a system of inertial principal axes, then

Hjk = diag[hkj] (3.7)

where

hkj =

∫

mk

kx2kj dmk j =1,2,3 hk4 =

∫

mk

dmk = mk

Eventually, the kinetic energy of the whole system is calculated as follows

T =
1

2

m
∑

k=1

tr(Ḃ
(b)
k HbkḂ

(b)⊤
k )+

1

2

n
∑

k=1

tr(Ḃ
(f)
k HfkḂ

(f)⊤
k )

(3.8)

Ḃ
(b)
k =

dB
(b)
k

dt
=
m
∑

i=1

∂B
(b)
k

∂q
(b)
i

q̇
(b)
i =

m
∑

i=1

B
(b)
ki
q̇
(b)
i

and

B
(b)
ki
=
∂B
(b)
k

∂q
(b)
i

Ḃ
(f)
k =

dB
(f)
k

dt
=
d(B0B

′(f)
k
)

dt
=
dB0
dt
B
′(f)
k
+B0

dB
′(f)
k
)

dt
= (3.9)

=
m
∑

i=1

B
(b)
0i B

′(f)
k

q̇
(b)
i +B0

6
∑

i=1

∂B
′(f)
k

∂q
(f)
ki

q̇
(f)
ki
=
m
∑

i=1

B
(b)
0i B

′(f)
k

q̇
(b)
i +B0

6
∑

i=1

B
′(f)
ki

q̇
(f)
ki

Having calculated the necessary derivative and after necessary transforma-
tions, the Lagrange equation component dependent on the kinetic energy for
the rigid part can be written in the following form (for j =1, ...,m)

ε
(b)
j =

d

dt

∂T(b)

∂q̇
(b)
j

−
∂T(b)

∂q
(b)
j

=
m
∑

i=1

a
(b)
ji q̈
(b)
k
+
n
∑

k=1

6
∑

i=1

a
(f)
jki
q̈
(f)
ki
+

(3.10)

+
m
∑

i=1

m
∑

l=1

b
(b)
jil
q̇
(b)
i q̇
(b)
l
+2

m
∑

i=1

n
∑

k=1

6
∑

l=1

b
(bf)
jikl

q̇
(b)
i q̇
(f)
kl

where

a
(b)
ji =

m
∑

k=1

tr(B
(b)
kj
HbkB

(b)⊤
ki
)+

n
∑

k=1

tr
(

B0jB
′(f)
k
Hfk(B0iB

′(f)
k
)⊤
)



Interactions between vibrations of... 455

a
(f)
jki
= tr(B0jB

′(f)
k
HfkB

(f)⊤
ki
)

b
(b)
jil
=
m
∑

k=1

tr(B
(b)
kj
HbkB

(b)⊤
kil
)+

n
∑

k=1

tr
(

B0jB
′(f)
k
Hfk(B0ilB

′(f)
k
)⊤
)

b
(bf)
jikl
= tr

(

B0jB
′(f)
k
Hfk(B0iB

′(f)
kl
)⊤
)

And the component for the flexible part equals (for k =1, ...,n; j =1, ...,6)

ε
(f)
kj
=

d

dt

∂T(f)

∂q̇
(f)
kj

−
∂T(f)

∂q
(f)
kj

=
m
∑

i=1

a
(b)
kji
q̈
(b)
i +

6
∑

i=1

a
(f)
kji
q̈
(f)
ki
+

(3.11)

+
m
∑

i=1

m
∑

l=1

b
(b)
kjil

q̇
(b)
i q̇
(b)
l
+2

m
∑

i=1

6
∑

l=1

b
(bf)
kjil

q̇
(b)
i q̇
(f)
kl

where

a
(b)
kji
= tr

(

B
(f)
kj
Hfk(B

′

0iB
(f)
k
)⊤
)

a
(f)
kji
= tr

(

B
(f)
kj
Hfk(B0B

′(f)
ki
)⊤
)

b
(b)
kjil
= tr

(

B
(f)
kj
Hfk(B0ilB

′(f)
k
)⊤
)

b
(bf)
kjil
= tr

(

B
(f)
kj
Hfk(B0iB

′(f)
kl
)⊤
)

The potential energy of the system consists of the energy of spring defor-
mation of the sdes and the energy of gravity forces.When the sdee connects
the elements l and p (Fig.3), then the elastic energy of this element can be
formulated as follows

Vse =
1

2

3
∑

j=1

cej(x
e
pj −x

e
lj)
2+
1

2

3
∑

j=1

kej(ϕ
e
pj −ϕ

e
lj)
2 (3.12)

where
cej – coefficients of longitudinal stiffness (Kruszewski et al., 1975)

kej – coefficients of rotational stiffness (Kruszewski et al., 1975)

and xelj, x
e
pj, ϕ

e
lj, ϕ

e
pj are the co-ordinates with respect to the base system.



456 I.Adamiec-Wójcik

Fig. 3. sde element connecting two rfes

The potential energy of the whole system is a sum of the elastic strain
energies of all sdes

Vs =
nest
∑

e=1

Vse (3.13)

In order to differentiate the expression for spring deformation energy it is
assumed that the co-ordinates of the sdee with respect to the co-ordinate
systems connected with rfel and rfep are defined by the following vectors,
respectively

l
re = [ηl1,ηl2,ηl3]

⊤ in the {l} system

p
re = [ηp1,ηp2,ηp3]

⊤ in the {p} system
(3.14)

The elastic energy of the sdee is definedwith respect to the co-ordinate system
{r} connected with the moving base. Thus, the following holds

r
rel =

r
l′T
l′

lT
l
re

r
rep =

r
p′T
p′

pT
p
re (3.15)

where
r
l′T,

r
p′T – matrices with constant coefficients

l′

lT,
p′

pT – matrices with coefficients dependent on the gene-
ralised co-ordinates of rfel and rfep, respectively.

Thematrices i
′

iT for i ∈{l,p} are defined in (2.4).

After necessary calculations we obtain the following formulae for the co-
ordinates

xei1 = ci2ci3ηi1−ci2si3ηi2+si2ηi3+xi1+ui1
(3.16)

xei2 =(si1si2ci3+ ci1si3)ηi1+(ci1ci3−si1si2si3)ηi2−si1ci2ηi3+xi2+ui2

xei3 =(si1si3− ci1si2ci3)ηi1+(ci1si2si3+si1ci3)ηi2+ ci1ci2ηi3+xi3+ui3



Interactions between vibrations of... 457

where ui = [ui1,ui2,ui3]
⊤ is the position vector of the co-ordinate system

connected with the ith rfe with respect to the co-ordinate system {r}.

Having used (3.12) the derivatives of the elastic energywith respect to the
generalised co-ordinates can be written as follows

∂Ve
∂xpj

= cej(x
e
pj −x

e
lj)

∂Ve
∂ϕpj

= kej(ϕ
e
pj −ϕ

e
lj)+

3
∑

k=1

cek(x
e
pj −x

e
lj)
∂xepk
∂ϕepj

(3.17)

∂Ve
∂xlj
=−cej(x

e
pj −x

e
lj)

∂Ve
∂ϕlj

=−kej(ϕ
e
pj −ϕ

e
lj)−

3
∑

k=1

cek(x
e
pj −x

e
lj)
∂xelk
∂ϕe
lj

The potential energy of the gravity forces can be calculated as follows

Vp = gΘ
⊤

3

(

m
∑

i=1

m
(b)
i B

(b)
i rbci+

n
∑

i=1

mfiBiΘ4
)

(3.18)

where
mbi – mass of the ith link of the base system
mfi – mass of the ith rfe of the flexible link

and

Θ4 = [0,0,0,1]
⊤

Θ
⊤

3 = [0,0,1,0]

The respective derivatives are calculated as follows

∂Vp

∂q
(b)
j

= gΘ⊤3

(

m
∑

i=1

mbiB
(b)
ij rbci+

n
∑

i=1

mfiB0jB
′

iΘ4

)

j =1, ...,m

∂Vp

∂q
(f)
kj

= gΘ⊤3 mfkB
(f)
kj
Θ4

k =1, ...,n
j =1, ...,6

(3.19)

Substituting equations (3.11), (3.17) and (3.19) into the Lagrange equ-
ations (3.1), the equations of motion, which form a set of m+6n non-linear
differential equations of the second order, are obtained. Various methods can
be used for integrating those equations.



458 I.Adamiec-Wójcik

4. Example of calculations

The derived equations of motion can be very easily adapted for dynamic
analysis of a systemwhenthebasemotion is known.Numerical calculationsha-
ve been carried out for amanipulator described byKane et al. (1987) (Fig.4).

Fig. 4. Model of the manipulator

Kane et al. (1987) considered an example of base motion called a deploy-
ment process, which is characterised by a smooth change of the angles

ψi(t)=















π−
π

2
αi
(

t−
T

2π
sin
2πt

T

)

for 0¬ t ¬ T

π

2
for t > T

(4.1)

where i =1,2,3, α1 =1/2, α2 =3/4, α3 =1 and T =15.

Wittbrodt andWojciech (1995) as well as Płosa andWojciech (2000) pre-
sented a comparison of results obtained using the RFEMwith those obtained
using the FEM and presented by Kane et al. (1987) and Du et al. (1992).
A very good correspondence of the results has been achieved when the base
motion was known.

For the purpose of this paper it is assumed that the base motion is not
knownand the casewhen each rfe has only threedegrees of freedom in relative



Interactions between vibrations of... 459

motion (xi1 = xi2 = xi3 = 0) is considered. The motion of the system is
realised by application of external forces, and the following procedure has
been applied:

1. Mass parameters are attributed to links 1 and 2 of the manipulator.
It is assumed that E = 6.895 · 1010Nm−2, G = 2.6519 · 1010Nm−2,
ρ =2766.67kgm−3 andboth links have a square box cross-sectionwhere
hz = 0.05m and hw = 0.0475m are external and internal dimensions,
respectively. The third link has the same parameters as in the above
mentioned papers and consists of two segments. The first segment B1
is 2.667m long and has a symmetric box cross section, while the second
segment B2 is a 5.333m long channel. Both segments are made of a
material for which E = 6.895 · 1010Nm−2, G = 2.6519 · 1010Nm−2,
ρ = 2766.67kgm−3. The section parameters for B1 are: cross-section
area A =3.84 ·10−4m2, cross-sectional moments of inertia Iyy = Izz =
1.5·10−7m4, the torsional oneabout theaxis J =2.2·10−11m4, sectional
mass centre offset xgc = 0, while the corresponding parameters of B2
are: A = 7.3 ·10−5m2, Iyy = 8.2181 ·10

−9m4, Izz = 4.8746 ·10
−9m4,

J =2.433 ·10−7m4, xgc =0.01875m.

2. Torquesnecessary for realisation of themotiondescribedby (4.1) are cal-
culated assuming that all three links are rigid. The equations of motion
of the rigid system can be obtained as a particular case of the general
equations of motion assuming that the number of rfes=0.

3. The torques calculated in 2havebeenapplied to the joints assuming that
the third link is flexible and is divided into 6 rfes. Having integrated
the equations of motion the angles ψi, i = 1,2,3, their velocities and
accelerations as well as deflections and rotation of the tip of the flexible
link have been calculated.

In the rest of the paper the kinematic input means the case of motion
realised by a change in the angles described by (4.1), and the force input
means the torques, calculated according to the above procedure, applied to
the joints.

Figure 5 shows a comparison of the deflections and rotation of the tip of
the flexible link obtained for the kinematic and force input. The denotations
by Kane et al. (1987) are used and the following holds

u1 =−q
(f)
n,1−

1

2
q
(f)
n,5ln u2 =−q

(f)
n,2−

1

2
q
(f)
n,4ln θ = q

(f)
n,6



460 I.Adamiec-Wójcik

Fig. 5. Deflections u1, u2 and rotation θ of the end of the third link; 1 – kinematic
input, 2 – force input

Fig. 6. Torque in the third joint required for realisation of motion; 1 – rigid link,
2 – flexible link



Interactions between vibrations of... 461

where q
(f)
n,i , i =1, ...,6 are the generalised co-ordinates of the rfen, and ln is

its length.

The torquewhich shouldbeapplied to joint 3 in order to ensure themotion
as in (4.1) is presented in Fig.6. One course is calculated assuming that the
third link is rigid and the other one when the link is flexible.

It canbeseen that thedifference is quite large,whichmeans that the torque
calculated for a rigid system can be applied to a flexible system only when
small deflections are considered. However, the procedure presented enables us
to estimate whether the influence of the flexible vibrations on the basemotion
is significant.

Flexibility also affects the velocities andaccelerations of the joint variables.
The influence of vibrations of the third link on the first and second joints is
not as large as on the third one.

Fig. 7. Velocity in the third joint; 1 – kinematic input, 2 – force input

Fig. 8. Accelerations in the first and second joint, respectively; 1 – kinematic input,
2 – force input



462 I.Adamiec-Wójcik

Figure 7 presents the velocity in the third joint for both kinematic and
force inputs, while Fig.8 shows the accelerations in the first and second joints
for the same inputs.

5. Final remarks

The paper presents the rigid finite element method applied to dynamic
analysis of systems with flexible links when the basemotion is not known.An
essential feature of the method is that the equations of motion are derived
only once, and they allow us to analyse both rigid and flexible systems, the
base motion is known or unknown. This means that not only the influence of
the base motion on the vibrations of a flexible link can be analysed but also
the effect which flexibility has on the base motion.

Themethod can be easily generalised to consider more complex problems
when the flexible links are found between chains of rigid links.

The aim of the paper is to show that in certain cases it is necessary to
take into account the influence of flexible vibrations on the basemotion. This
influence depends on the parameters of links realising the basemotion as well
as on the drive input.

References

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Wzajemne oddziaływanie ruchu bazowego i drgań członów podatnych

manipulatorów

Streszczenie

W artykule przedstawiono zastosowaniemetody Sztywnych Elementów Skończo-
nych do modelowania członów podatnych manipulatora. Równania ruchu wyprowa-
dzono zakładając, że ruch bazowy nie jest znany, czyli, że zależy od pewnych współ-
rzędnych uogólnionych. Przeprowadzone obliczenia numeryczne pokazują wzajemne
oddziaływanie drgań członu podatnego oraz ruchu bazowego.

Manuscript received March 1, 2001; accepted for print May 16, 2001