Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 1, pp. 51-68, Warsaw 2008

ON THE ANALYTICAL SOLUTION FOR THE SLEWING
FLEXIBLE BEAM-LIKE SYSTEM: ANALYSIS OF THE
LINEAR PART OF THE PERTURBED PROBLEM

André Fenili

National Institute for Space Research (INPE), São José dos Campos, Brazil

University of Taubaté, Department of Engineering Mechanics, Taubaté, SP, Brazil

e-mail: fenili@dem.inpe.br; fenili@unitau.br

José Manoel Balthazar

Departamento de Estat́ıstica Matemática Aplicada e Computação, UNESP, Rio Claro, SP, Brazil

Mechanical Design Department, UNICAMP, Campinas, SP, Brazil

e-mail: jmbaltha@rc.unesp.br

The dynamical system investigated in this work is a nonlinear flexible
beam-like structure in slewing motion. Non-dimensional and perturbed
governing equations ofmotion are presented. The analytical solution for
the linear part of these perturbed equations for ideal and for non-ideal
cases are obtained. This solution is necessary for the investigation of
the complete weak nonlinear problemwhere all nonlinearities are small
perturbations around a linear known solution. This investigation shall
help the analyst in the modelling of dynamical systems with structure-
actuator interactions.

Key words: non-ideal system, flexible structure, perturbed equations,
analytical solution

1. Introduction

The study of dynamic behaviour (and control) of slewing flexible structures
has in view the improvement of lightweight and faster structures. These in-
vestigations are complex and present continuing interest from researchers and
scientists.



52 A. Fenili, J.M. Balthazar

The applications of the theory of slewing flexible structures may be divi-
ded basically into two groups: robotics and aerospace structure applications.
Typically, themodal analysis approach is themost popularmodel approach to
aerospace slewing structures, and the finite element approach is the onemost
frequently used to investigate robotic manipulators. Low inherent damping,
small natural frequencies, and extreme light weights are some common cha-
racteristics of these systems, and which make them vulnerable to any exter-
nal/internal disturbances (such as angular maneuvers, impacts, etc). Robot
arms with such characteristics are easy to carry out, need smaller actuators
and can reach objectives in a greaterworkspace since they are thinner and lon-
ger than the rigid ones usually used for the same task. DCmotors are popular
actuators for lightweight manipulators not only because they can generate a
wide range of torque and angular velocity, but also because they are quiet,
clean and efficient. Nowadays, in the competitive world, the search for such
kind of mechanical systems is an increasing preoccupation.

Many of the publishedpapers in this area are concernedwith the dynamics
and/or control of flexible beam-like structures. Little effort has been focused
on the actuator-structure interaction. This interaction affects the dynamics of
the whole slewing system (actuator included).

The goal of this work is to deal with this interaction. In systems such
as lightweight robotic manipulators, solar panels and antennas in satellites,
helicopter blades and so on, the mutual interaction between the angular di-
splacement of the slewing axis and the flexible structure deflection can be very
important inhighangular speedmaneuvers (Balthazar et al., 1999, 2001-2004).

Dynamical systems, in which this interaction occurs, are called non-ideal
systems (Kononenko, 1969). In the ideal system, there is nomutual interaction
and only the actuator dynamics excites the structure dynamics.

Governing equations of motion for the ideal and non-ideal damped beam-
like slewing flexible structure connected to a DC motor are presented with
discussions on nonlinear effects in Fenili (2004), Fenili et al. (2004). These
equations and boundary conditions are nondimensional and scaled quantities
also known as perturbed equations of motion because of the small parameter
which multiplies nonlinear and damping terms (Fenili, 2000; Fenili and Bal-
thazar, 2005a,b). Some experimental results and discussions were related in
Fenili et al. (2001).

Next, some papers are mentioned from the current literature concerning
the subject treated here, in order to understand better the position of this
problem in themain current literature.



On the analytical solution for the slewing... 53

2. A brief literature review

Slewing flexible structures were first considered, in literature, by Book et al.
(1975). In that paper, the authors applied amodal truncatedmodel on a sys-
tem composed by two flexible beams with two joints, and included discussion
on the controller design with a torque source. Considerable efforts have been
performed since then by other authors in this direction. Some of those authors
arementioned next (without undeservingmany others notmentioned) in this
paper.

Cannon and Schmidt (1984) developed and demonstrated stable and pre-
cise position control of one end of a very flexible beam by using direct me-
asurements from the free end position as a basis for torquing at the other
end. The authors demonstrated that a satisfactory feedback tip-control re-
sponse can be achieved with a good dynamic model of a flexible arm. Bayo
(1987) analyzed a structural finite element algorithm on the linear Bernoulli-
Euler beam in order to calculate the end torque necessary to produce desired
motion at the free tip of a flexible link. The computed torque, smaller than
the one required for the rigid link of some weight, provided desired tip mo-
tion without overshoot. The authors used this approach in both open loop
and feedback controls. Juang et al. (1986) studied a slewing flexible structu-
re experimentally; Juang and Horta (1987) developed a hardware set-up to
study slewing control for slewing flexible structures and using linear optimal
control to design active control (implemented in an analog computer); Juang
et al. (1989) discussed several important issues related to slewing experiments
with flexible structures including nonlinearities and calibration of actuators
and sensors. In Hamilton et al. (1991) a simple experimental set-up wasmade
up by a pair of single-axis flexible beams attached to a DC servo motor in
order to illustrate collocated and noncollated control for this kind of struc-
tures. An optical encoder and strain gauges provided hub and beam position
information, respectively. Their results on non-collocated zero-placement con-
trol illustrate that the feedback from the beam and motor hub provides the
necessary information for vibration suppression in slewing maneuvers. Yang
et al.) (1994) realized numerical simulations concerning slewing control tasks
for a planar articulated double-beam structure. Garcia and Inman (1990) in-
vestigated the modeling of a single link flexible slewing beam torque driven
by a DC motor at the slewing axis. Effects of modal participation factors in
the slewing equations of motion were discussed. These factors were indicative
of the degree into which actuators and flexible structures interacted with one
another dynamically. Boundary conditions of the slewing beamwere determi-



54 A. Fenili, J.M. Balthazar

ned by the actuator that drove the entire system. Garcia and Inman (1991)
considered the slewing control of an active flexible structure by examining
the governing equations of motion of an integrated actuator-structure system
composed by a thin aluminum beam torque driven by an armature controlled
electric motor and actuated by a piecewise distributed piezoceramic actuator.
Their approach offered an advantage of reducing the peak voltage demands
on the motor. In addition, that active structure approach substantially redu-
ced the maximum tip deflection of the beam. Sah et al. (1993), using a finite
element algorithm, considered the closed-loop performance and the dynamic
interaction between a DC motor and a slewing beam. The authors conclu-
ded that systems with an appropriate amount of actuator-beam interaction
tend to be more easily controlled and require a modest amount of actuator
efforts, but systems with little actuator-beam interaction are especially prone
to beam vibrations and require feedback of beam dynamics for good closed-
loop performance. It was also shown that systems with excessive interaction
require stabilization efforts in order to obtain good transient performance and
tend to consume increased levels of actuation energy. In Kwak et al. (1994),
a slewing flexible beam equippedwith piezoelectric sensors and actuators was
modeled via extended Hamilton’s principle considering nonlinearities coming
from rigid body rotational motion, whereas the elastic vibration was assumed
to be small compared to rigid body rotation. The authors used a decentralized
control technique in which the control of the whole system was divided into
the slewing control and the vibration suppression control. The sliding model
control was proposed as the slewing control, and themodal space positive fe-
edback plus disturbance-counteracting control was developed for suppression
of vibrations. These techniques were verified by experiments. The experimen-
tal results showed that the decentralized control performs satisfactorily, but
high frequency vibrations remained uncontrolled. In this case, there will be a
need for more actuators, sensors andmore powerful piezoelectric actuators.

Finally, Fenili (2000) has started the studyof nonlinear dynamicbehaviour
of flexible structures in planar slewingmotions for linear and nonlinear curva-
ture assumptions. A schematic of the system investigated there (and here) is
depicted in Fig.1. In this paper, the authors present a search for an approxi-
mate analytical solution to this problem. A short version of this subject was
discussed in Fenili and Balthazar (2005a). An experimental apparatus was
constructed in order to verify the theoretical developments and is shown in
Fig.2 (Fenili et al., 2001).

This paper is organized as follows. In Section 3 the governing equations of
motion are presented. In Section 4 the analytical solution of the linear non-



On the analytical solution for the slewing... 55

Fig. 1. Schematic of a slewing beam-like flexible structure (Fenili, 2000)

Fig. 2. Experimental set-up of the slewing flexible beam-like structure (Fenili, 2000;
Fenili et al., 2001)

ideal problem(onemodeexpansion) isdeveloped. InSection5 someconcluding
remarks are presented. In Section 6 someacknowledgements aremade. Finally,
main bibliographic references used in this work and two appendixes in order
to detail and/or give values for the expressions, equations and coefficients
presented along the paper are listed.

3. Governing equations of motion

Thegoverning equations ofmotion for the systemdepicted inFig.1 are derived
from the extended Hamilton principle (Fenili, 2000; Fenili et al., 2004a). This
mathematical model is based on a nonlinear curvature assumption for the
beam. The nonlinear equations obtained are put on a dimensionless form and



56 A. Fenili, J.M. Balthazar

scaled in order to become perturbed equations of the kind (Nayfeh andMook,
1973)

linear terms + ǫ2(nonlinear terms + damping) = 0 (3.1)

The small parameter, ǫ, in (3.1) is given by

ǫ=
r2

L2
(3.2)

where r is the radius of gyration of the cross-section area of the beamand L is
the length of the beam.The perturbed equations are then discretized through
the assumed method of modes. On this assumption, the deflection variable,
v(x,t), is written as

v(x,t) =
n
∑

i=1

φi(x)qi(t) (3.3)

where n is the number of modes, φi(x) is any admissible function which
represents the assumed solutions in space and qi(t) are unknown solutions in
time. Considering the longitudinal deflection of the beam, u(x,t), as being
of the order O(ǫ2), assuming θ as the slewing angle and ia as the armature
current in the DC motor, the perturbed discretized governing equations of
motion for thenon-ideal system,after somemanipulations inorder to eliminate
u(x,t) from the set of governing equations (Fenili, 2000), are given by

i̇a+ c1ia+ c2θ̇= c1U

θ̈+ c3θ̇− c4ia− c5φ
′′

1(0)q1 =0 (3.4)

q̈1+w
2
1q1+α1θ̈+ ǫ

2[µq̇1+β11θ̇
2q1−℘111θ̇q1q̇1+λ111θ̈q

2
1 +

+Λ1111q1q̇
2
1 +Λ1111q

2
1q̈1+Γ1111q

3
1] = 0

The boundary conditions are given by

φ(0, t) = 0 φ′(0, t) = 0

φ′′(1, t) = 0 φ′′′(1, t) = 0

The coefficients of Eqs. (3.4) are presented inAppendixA. For the linear ideal
problem, one must consider c5 =0 in Eq. (3.4)2.

4. Analytical solution to the linear problem (one mode
expansion)

For the linear system (ǫ = 0 in Eq. (3.4)3) associated with the primary re-
sonance of the first flexural mode, one needs to make U = 0 in Eq. (3.4)1.



On the analytical solution for the slewing... 57

To run numerical simulations, the initial condition is given in the variable q1
(associated to the time behaviour of the beam deflection), as can be seen in
Figures 3 to 5.
Considering one mode expansion (n=1), the set of equations to be ana-

lytically solved is given by

i̇a+ c1ia+ c2θ̇=0

θ̈+ c3θ̇− c4ia− c5φ
′′

1(0)q1 =0 (4.1)

q̈1+w
2
1q1+α1θ̈=0

Themode shapes considered, are given by

φi(x)= cosh(aix)− cos(aix)−αi[sinh(aix)− sin(aix)]

where

αi =
cosh(aiL)+cos(aiL)

sinh(aiL)+sin(aiL)

and ai is associated with the eigenvalues of the clamped-free undamped and
not excited linear (Bernoulli-Euler) beam.

4.1. Analytical solution for q1(t)

Dividing Eq. (4.1)3 by −α1, adding to Eq. (4.1)2 and solving for ia, gives

ia =−
( 1

α1c4

)

q̈1−
( w21
α1c4

+
c5φ
′′

i (0)

c4

)

q1+
(c3

c4

)

θ̇ (4.2)

Differentiating (4.2) with respect to time, yields

i̇a =−
( 1

α1c4

)...
q 1−

( w21
α1c4

+
c5φ
′′

i (0)

c4

)

q̇1+
(c3

c4

)

θ̈ (4.3)

Substituting Eqs. (4.2) and (4.3) into (4.1)1, results

−

( 1

α1c4

)...
q 1−

( c1

α1c4

)

q̈1−
( w21
α1c4

+
c5φ
′′

i (0)

c4

)

q̇1−
(c1w

2
1

α1c4
+
c1c5φ

′′

i (0)

c4

)

q1+

(4.4)

+
(c3

c4

)

θ̈+
(c1c3

c4
+c2
)

θ̇=0

Solving Eq. (4.1)3 for θ̈, gives

θ̈=−
( 1

α1

)

q̈1−
(w21
α1

)

q1 (4.5)



58 A. Fenili, J.M. Balthazar

Integrating Eq. (4.5) from 0 to t, are obtains:

θ̇=−
( 1

α1

)

q̇1−
(w21
α1

)

t
∫

0

q1 dt (4.6)

SubstitutingEqs. (4.1)1,2 into (4.4), differentiating the resulting equationwith
respect to time andmultiplying by −1, results

A
....
q 1+B1+Cq̈1+Dq̇1+Eq1 =0 (4.7)

where:

A=
1

α1c4
B=
c1+c3
α1c4

C =
w21 + c1c3
α1c4

+
c5φ
′′

1(0)

c4
+
c2

α1
D=

(c1+ c3)w
2
1

α1c4
+
c1c5φ

′′

1(0)

c4

E =
c1c3w

2
1

α1c4
+
c2w
2
1

α1

Equation (4.7) is a fourth order ordinary differential equation with constant
coefficients in the variable q1. The characteristic equation associated to (4.7)
is given by

Ar4+Br3+Cr2+Dr+E=0 (4.8)

If Eq. (4.8) has only real and not equal roots, a solution to (4.7) is of the form

q1 =C1e
r1at+C2e

r2at+C3e
r3at+C4e

r4at (4.9)

If Eq. (4.8) has only complex roots, the solution to (4.7) is of the form

q1 =C1e
r1atcos(r1bt)+C2e

r2at sin(r2bt)+C3e
r3atcos(r3bt)+C4e

r4at sin(r4bt)
(4.10)

If Eq. (4.8) has two complex and two real roots, the solution to (4.7) is of the
form

q1 =C1e
r1atcos(r1bt)+C2e

r2at sin(r2bt)+C3e
r3at+C4e

r4at (4.11)

where ria and rib are associated which each of the possible complex roots of
(4.8).
Let the initial time be represented by tr. Equations (4.1) can be rewritten

in tr, resulting

i̇ar =−c1iar − c2θ̇r

θ̈r =−c3θ̇r+ c4iar + c5φ
′′

i (0)q1r (4.12)

q̈1r =−w
2
1q1r−α1θ̈r



On the analytical solution for the slewing... 59

Differentiating (4.12)2,3 with respect to time, one finds
...
θ r and

...
q 1r.

Using Eqs. (4.9), (4.10) or (4.11) and their time derivatives (in t= tr), a
set of algebraic equations can be formulated. From these equations, one can
find the constants Ci. With the values in Appendix B, one can easily find
A, B, C, D and E, and calculate roots of the characteristic equations. The
solution given by Eq. (4.11), for each case, is given by

q1case1 =1.1958 ·10
4e−24.7510t cos(962.4100t)+345.9672e−24.7510t ·

·sin(962.4100t)+60.3107e−613.4746t +2.0655e−6.4628·10
−5t

q1case2 =−11.4037e
−47.1299t cos(78.5155t)+4.9359e−47.1299t sin(78.5155t)−

−0.0433e−592.6730t +8.4131 ·104e−0.0113t (4.13)

q1case3 =7.8895 ·10
5e−0.5874t cos(0.8169t)−6.3616 ·105e−0.5874t sin(0.8169t)+

+15.2815e−591.4816t −0.1039 ·105e−95.4528t

Fig. 3. Non-dimensional transverse deflection of the versus time – case 1

Fig. 4. Non-dimensional transverse deflection of the versus time – case 2

Figures 3 to 5 illustrate solutions (4.13). These solutions are stable and
damped.



60 A. Fenili, J.M. Balthazar

Fig. 5. Non-dimensional transverse deflection of the versus time – case 3

5. Analytical solution for θ(t)

Integrating Eq. (4.6) in time and using solution (4.11), are obtains

θ=E1e
r1atcos(r1bt)+E2e

r1at sin(r1bt)+E3e
r2atcos(r2bt)+

(5.1)

+E4e
r2at sin(r2bt)+E5e

r3at+E6e
r4at+E7t+E8

where

E0 =
C1w

2
1(r
2
1a−r

2
1b)

α1(r
2
1a+r

2
1b
)2

E1 =−
C1+E0α1
α1

E2 =−
2C1w

2
1r1ar1b

α1(r
2
1a+r

2
1b
)2

E3 =−
2C2w

2
1r2ar2b

α1(r
2
2a+r

2
2b
)2

E4 =−
C2((r

2
2a+r

2
2b)
2+w21(r

2
2a−r

2
2b))

α1(r
2
2a+r

2
2b
)2

E5 =−
C3(r

2
3a+w

2
1)

α1r
2
3a

E6 =−
C4(r

2
4a+w

2
1)

α1r
2
4a

E7 =
w21
α1

( C1r1a

r21a+r
2
1b

+
C3

r3a
+
C4

r4a
−

C2r2b

r22a+r
2
2b

)

E8 =
w21
α1

(C3

r23a
+
C4

r24a
+
E3α1

w21

)

+E0

Figures 6 to 8 show solution (4.15).
For case 1, the angular displacement is damped. For cases 2 and 3, the

oscillation of the beam is damped but the angular displacement increases in
time.



On the analytical solution for the slewing... 61

Fig. 6. Angular displacement (nondimensional) – case 1

Fig. 7. Angular displacement (nondimensional) – case 2

Fig. 8. Angular displacement (nondimensional) – case 3

5.1. Analytical solution for i
a
(t)

Substituting q1 (Eq. (4.11)), its second timederivative and θ̇ (usingEq. (4.2)),
gives

ia =(H1cos(r1bt)+H2 sin(r1bt))e
r1at+[H3 sin(r2bt)+H4cos(r2b)]e

r2at+
(5.2)

+H5e
r3at+H6e

r4at+H7+H8



62 A. Fenili, J.M. Balthazar

where

H1 =−
C1(r

2
1a−r

2
1b+w

2
1)

α1c4
−

C1c50+ c3A(E1r1a+E2r1b)

c4

c50 = c5φ
′′

i (0) H2 =
2C1r1ar1b
α1c4

−H2b

H2b =
c3A(E1r1b−E2r1a)

c4

H3 =−
C2(r

2
2a−r

2
2b+w

2
1)

α1c4
−

C2c50− c3B(E3r2b−E4r2a)

c4

H4 =−
2C2r2ar2b
α1c4

+
c3B(E3r2a+E4r2b)

c4

H5 =H5b−
C3c50+ c3E5r3a

c4

H5b =−
C3(r

2
3a+w

2
1)

α1c4

H6 =−
C4(r

2
4a+w

2
1)

α1c4
−

C4c50+ c3E6r4a
c4

H7 =
c3E7

c4

H8 =−H1−H4−H5−H6−H7

Fig. 9. Armature current (nondimensional) – case 1

The constant H8 is introduced in order to guarantee that ia(0) = 0. Fi-
gures 9 to 11 illustrate the solution given by Eq. (4.16) to each of the studied
cases.



On the analytical solution for the slewing... 63

Fig. 10. Armature current (nondimensional) – case 2

Fig. 11. Armature current (nondimensional) – case 3

6. Conclusions

The analytical solution to the set of differential equations associated with the
linear part of the perturbed governing equations of motion for the general
problem under investigation (nonlinear slewing flexible beam-like structures)
is presented here. This solution, considered of the order ǫ0, is very important
to start the search for a perturbed solution to the weak nonlinear problem
presented inEqs. (3.4). The analytical solutions obtainedhere for the actuator
and for the beam (and plotted in Fig. 3 to 11) are stable.

The next step in this investigation consists in applying a perturbation
technique to solve the perturbed problem using the solution presented here (a
remarkable task!). Using themultiple scalemethod, for example, the nonlinear
perturbed equations of motion of the order ǫ2 to be solved now are given by

∂ia0

∂T0
+ c1ia0+ c2

∂θ0

∂T0
+ ǫ2
[∂ia1

∂T0
+
∂ia0

∂T1
+ c1ia1+c2

∂θ1

∂T0
+ c2
∂θ0

∂T1

]

=

= ǫ2
[uc1

2i
(eiΩt− e−iΩt)

]



64 A. Fenili, J.M. Balthazar

∂2θ0

∂T20
+ c3
∂θ0

∂T0
− c4ia0− c5φ

′′

1(0)q10+

+ǫ2
[∂2θ1

∂T20
+ c3
∂θ1

∂T0
− c4ia1− c5φ

′′

1(0)q11+ c3
∂θ0

∂T1
+2

∂θ0

∂T0∂T1

]

=0

∂2q10

∂T20
+w21q10+α1

∂2θ0

∂T20
+ ǫ2
[∂2q11

∂T20
+w21q11+α1

∂2θ1

∂T20
+β11q10

(∂θ0

∂T0

)2

+

−℘111q10
∂θ0

∂T0

∂q10

∂T0
+λ111q

2
10

∂2θ0

∂T20
+Λ1111q10

(∂q10

∂T0

)2

+

+Λ1111q
2
10

∂2q10

∂T20
+Γ1111q

3
10+2α1

∂θ0

∂T0∂T1
+2

∂q10

∂T0∂T1

]

=0

A. The coefficients of equations (3.4)

c1 =
RaT

La
c2 =Ng c3 =

cvN
2
gT

It

c4 =
NgKtKbT

2

LmIt
c5 =

EIT2

LIt
It = Ishaft +N

2
gImotor

Rij =

x
∫

0

φ′i(ξ)φ
′

j(ξ) dξ=Rji Vi =−

1
∫

x

φi(ξ) dξ

Λijkℓ =

1
∫

0

(Sjkφ
′′

iφℓ+Rjkφ
′

iφℓ) dx αℓ =

1
∫

0

xφℓ dx

Sij =−

1
∫

x

[

η
∫

0

φ′i(ξ)φ
′

j(ξ) dξ
]

dη

Wij =−

1
∫

x

φ′i(ξ)φj(ξ) dξ

℘ijℓ =2

1
∫

0

(Rijφℓ−φ
′′

iVjφℓ−φ
′

iφjφℓ) dx

βiℓ =
[

1
∫

0

(

xφ′iφℓ+
1

2
(x2−1)φ′′iφℓ

)

dx
]

−1



On the analytical solution for the slewing... 65

λijℓ =

1
∫

0

(

−

1

2
Rijφℓ+φ

′′

iVjφℓ+φ
′

iφjφℓ

)

dx

Γijkℓ =

1
∫

0

[ 3

1.87804
φ′iφ
′′

jφ
′′′

kφℓ+
3

2(1.87804)
φ′′iφ

′′

jφ
′′φℓ+

+w2j(φ
′

iφjφ
′

kφℓ+Wijφ
′′

kφℓ)
]

dx

where: Ra represents the armature resistance, T is the period of the first
natural frequency of the beam, La – armature inductance, cv –motor internal
damping, Ishaft – inertia of the connectingmotor-beam shaft, Imotor – inertia
of the motor, Kt – torque constant, Kb – back e.m.f. constant, E – Young’s
modulus, I – inertia of the beam cross section around the neutral axis, L –
beam length, φℓ – each of the flexural vibrationmode and wℓ – corresponding
model frequency.

B. Beam properties (cases 1 to 3) and motor parameters

T1 =1 w1 =1 Ieixo =0.0000369kgm
2

α1 =0.570157 φ
′′

i (0)= 7.05377

B.1. Beam

Case 1: aluminum, L= 0.20m, E = 0.70 ·1011N/m2, height = 0.02544m,
height = 0.02544m, A = 0.00008039m2, I = 6.6896 · 10−11m4,
ρ=2700kg/m3, ǫ=0.000020804

Case 2: aluminum, L= 1.40m, E = 0.70 ·1011N/m2, height = 0.02544m,
basis = 0.00316m, A = 0.00008039m2, I = 6.6896 · 10−11m4,
ρ=2700kg/m3, ǫ=0.00000042456

Case 3: steel, L = 0.8720m, E = 2.10 · 1011N/m2, height = 0.01587m,
basis = 0.00082m, A = 0.00001301m2, I = 7.2918 · 10−13m4,
ρ=7800kg/m3, ǫ=0.000000073710

Obs.: ”height” and ”basis” refer to the beam cross section (Fenili,2000).



66 A. Fenili, J.M. Balthazar

B.2. DC motor parameters

Cm = 0.0046290Nms/rad, Kt = 0.0528140Nm/A, Ra = 1.9149520Ω,
Kb =0.0528140Vs/rad, Lm =0.0031000H, Imotor =0.0000654kgm

2.

Acknowledgements

The authors wish to thank Fundação de Amparo à Pesquisa do Estado de São

Paulo (FAPESP) and Conselho Nacional de Pesquisas (CNPq) for the grants that

supported this study.

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O analitycznym rozwiązaniu dla układu podatnej belki obrotowej: analiza
części liniowej zagadnienia perturbowanego

Streszczenie

Układem dynamicznym badanym w pracy jest podzespół zawierający podatną
belkę poruszającą się ruchem obrotowym w płaszczyźnie poziomej. Zaprezentowano
bezwymiarowe i perturbowane równania ruchu. Rozwiązania analityczne części zline-
aryzowanej tych równań uzyskano dla przypadku idealnego i nieidealnego. Taka po-
stać rozwiązania jest niezbędna do analizy zupełnego i słabo nieliniowego problemu,
w którym nieliniowości stanowią niewielkie perturbacje wokół rozwiązania liniowego.
Prezentowanewyniki badańmogą okazać się pomocą dla analityków zajmujących się
modelowaniem układów dynamicznych uwzględniających interakcje zachodzące po-
między daną konstrukcją bazową, a zamocowanymi na niej aktywnymi elementami
wykonawczymi (aktuatorami).

Manuscript received February 21, 2007; accepted for print July 16, 2007