Examined here are the effects of motor dynamics and secondary piezoceramic actuators on vibration suppression during the slewing of flexible structures. The approach focuses on the interaction between the structure, the actuators, and the choice of control law. The results presented here are all simulated, but are based on experimentally determined parameters for the motor, structure, piezoceramic actuators, and piezofilm sensors. The simulation results clearly illustrate that the choice of motor inertia relative to beam inertia makes a critical difference in the performance of the system. In addition, the use of secondary piezoelectric actuators reduces the load requirements on the motor and also reduces the overshoot of the tip deflection. The structures considered here are a beam and a frame. The majority of results are based on a Euler Bernoulli beam model. The slewing frame introduces substantial torsional modes and a more realistic model. The slewing frame results are incomplete and represent work in progress

Garcia, Ephrahim

Inman, Daniel J.

Pokines, Brett

English

NASA Technical Reports Server

wN91 "22339
Vibration Suppression and Slewing
Control of a Flexible Structure
Daniel J. Inmant
Ephrahim Garciatt
Brett Pokinesttt
Department of Mechanical & Aerospace Engineering
University at Buffalo
Buffalo, NY 14260
Abstract
This work examines the effects of motor dynamics and secondary piezoceramic actuators on
vibration suppression during the slewing of flexible structures. The approach focuses on the
interaction between the structure, the actuators and the choice of control law. The results
presented here are all simulated but are based on experimentally determined parameters for
the motor, structure, piezoceramics actuators and piezofilm sensors. The simulation results
clearly illustrate that the choice of motor inertia relative to beam inertia make a critical
difference in the performance of the system. In addition the use of secondary piezoelectric
actuators reduces the load requirements on the motor and also reduces the overshoot of the tip
deflection.
The structures considered here are a beam and a frame. The majority of the results are
based on an Euler Bernoulli beam model. The slewing frame introduces substantial torsional
modes and a more realistic model. The slewing frame results are incomplete and represent
work in progress.
1. Introduction
A slewing motion consists of the rotation of a structure about a point. In the case
considered here, a DC electric motor is used to move a beam and/or a frame about the axis of
the motor in order to orient the length of the structure in a new direction (see figure 1). In the
past, slewing maneuvers have been carried out on passive structures, i.e., structures
which have no internal control or sensing mechanisms. Here, the effects of slewing an active
structure are considered. An active or smart structure is defined as a structure with sensors
and actuators integrated within the structure (Wada, 1989). A passive structure does not
contain any integrated control hardware. The slewing of a passive beam has been considered
by several researchers. Garcia (1989) and Garcia and Inman (1990) examine the dynamic
interaction between the structure and actuator in slewing a passive beam. Juang et aI (1986),
Yurkovich and Tzes (1990), Cannon and Schmitz (1984) and Hastings and Book (1987)
have all consider the effects of slewing passive beams. Park et al (1989) considered slewing
a passive beam with a secondary voice coil actuator attached to improve vibration
suppression. Their results motivated the work presented here which considers the effects of
slewing an active beam. This presents a multiple input control problem. The active beana
consists of a flexible aluminum beam with embedded piezoelectric actuators and sensors.
The results for the active beam have been presented in preliminary form at the Army Research
t Professor and Chair
tt Visiting Assistant Professor
ttt Graduate Student
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Office Workshop (see Inman et al 1990). The slewing frame example is experimental and
represents work in progress.
The analysis proceeds by forming a Hamiltonian consisting of the elastic and kinetic
energy in the Euler-Bemoulli beam, plus the nonconservative work done on the beam by the
DC motor and piezoceramic actuator. Garcia (1989) illustrated that the dynamic interaction
between the slewing actuator, the DC motor, and the flexible structure can lead to improved
vibration suppression. Traditionally, the slewing control of a flexible single link structure has
been a single actuator problem. Park et al (1989) proposed the use of a "voice-coil" actuator
in addition to the slewing motor; this actuator was rigidly attached to the slewing hub and
actuated the beam near the clamped end. This approach achieved improved structural
dynamic performance and reduced peak motor voltages. However, these performance gains
were at the cost of adding the mass of the coil actuator and its supporting mechanical fixture
to the slewing payload.
We propose that direct structural actuation be achieved in the slewing maneuver by use
of a piezoceramic actuator. This active structure will contain a layered piece-wise distributed,
or segmented, piezoceramic crystal in the case of the beam and an active longeron element
consisting of bending piezoceramics for the frame. The active beam being considered here is
similar to those considered earlier in a damped configuration by Fanson and Caughey (1987)
and Burke and Hubbard (1987). Fanson and Caughey considered a cantilevered flexible
beam controlled by a collocated pair of piezoelectric actuators and strain sensors coupled with
a positive position feedback control law. In the case presented here a piezofilm will be used
instead of a piezoceramic for the strain measurement and piezoceramics are used for
actuators.
A theoretical optimal control study is performed using a linear quadratic regulator (LQR)
control formulation. This is presented only for the beam. A comparison of control laws is
made where the penalty function is varied to change the degree of control effort afforded by
the active beam. The goal here is to illustrate that increased vibration suppression may occur
in slewing maneuvers by taking advantage of control structure interaction and to investigate
the vibration suppression effects of slewing an active structure.
2. System Dynamics
The dynamics of the slewing beam system are developed from Hamilton's principle.
First, the dynamics of a slewing piezo-actuated structure are considered with the effects of a
piece-wise distributed piezo actuator. The actuator dynamics, that is, the interaction of motor
and beam are also modeled. The moment generated by the piece-wise distributed, piezo
actuator is calculated. Finally, the equations of motion for this active slewing structure are
assembled in a lumped mass model representation. The details can be found in Inman et al
(1990) and follws directly from Garcia (1989).
Figure 1 illustrates the coordinates used in defining the equations of motion of a flexible
structure undergoing a slewing motion through an angle 0(t). The deflection of the beana
y(x,t) is defined relative to the rigid motions 0. The torque causing the motion is denoted by
't. The beam, of length L, deforms and rotates in the X-Y plane. Figure 2 illustrates the
model of the motor. Figure 3 illustrates the use of piezoceramics in the slewing beam.
Yy(x,O _ be_n _I_._ x
Fig. 1. Slewing flexible beam schematic.
_X
O
1
ea
[
O
¢Wv 
La
(
Ra
.JNAI_A¢-.----
K b 8m 8'
'_a '_
Fig. 2. Motor armature circuit and gear box schematic.
EDGE VIEW
SERVO
AXIS
L4
Fig. 3. Schematic of the slewing beam showing the location of embedded
piezoceramic actuators.
3. Control Design
A linear quadratic regulator control law was designed to illustrate the effects of slewing
an active structure versus slewing a passive structure. The results are based on a
beam/motor system designed to take maximum advantage of the interaction between the
structural modes and the motor torque. Both the voltages to the electric motor and to the
embedded piezoceramic were used as control inputs.
To perform the control analysis, the mathematical model presented in Inman et al (1990)
is discretized in space and manipulated into state space form. Combining the governing
equations and assuming that only n terms are used in a modal approximation (5), the
equations of motion of the slewing active structure can be written in matrix form as the vector
differential equation
M_+ D Cl +K q =Bfu (1)
where the vector q is defined by qT = [0(t) ql(t) q2(t) ... qn(t)]. The mass, damping and
stiffness coefficient matrices are:
Ib+i s Ii+Isl-'l(O) ... In+IsFn(O) lM = [ II+IsF1 (0) M 1 i/
mln+IsFn(O) ... M n
(2)
D
I bv bvFl(O) "'" bvFn(0) l
bvI".l(0) bvFl(0)2 ... bvFl(0)Fn(0 )
•
mbvFn(0) bvFn(0)Fl(0).., bvFn(0)2
(3)
K
0 01xn _1
M1 _2
0nxl 1
... MIo_
(4)
where Fi = _i'(0), the ith modal participation factor. In addition, Mi is the ith modal mass, COl
denotes the structures natural frequencies and the ith inertia term is given by
L
Ii = Jpxq_i(x)dx
The control input vector u is the 2 x 1 vector tlT = [ea, Vp] and the control coefficient
matrix is
L 0 I_[_I(L1)-_(L2)] . _t[_n (L1) - _n (L2)]
(5)
for a single segment piezoceramic actuator.
Active control is performed by using state feedback and solving a standard linear
quadratic regulator (LQR) control law design. The system of Eq. (1) is first put into state
space form by defining the state vector x as
(6)
and the corresponding state matrix
A=[ 0 I
_MqK _M-l(
(7)
where 0 denotes the matrix of zeros and I denotes the identity matrix of appropriate
dimension. With this change of coordinates, Eq. (1) becomes
= Ax + Bu (8)
with output measurements defined by
y = Cx (9)
Here the matrix of constants C specifies which coordinates of the vector x are measured.
State feedback control is implemented by specifying the relation
u = - Kfx (10)
The LQR control algorithm then calculates the value of the gain matrix Kf such the cost
functional
J = ff(xrQx + uTRu)dt (11)
is minimized. The matrices Q and R are symmetric positive definite weighting matrices
which are chosen to produce acceptable responses. In the case presented here the matrix Q
was chosen to be
Q=diag[8 3 1 1 8 3 1 1] (12)
which places emphasis on minimizing the angular displacement (and velocity) and the first
modal displacement (and velocity). The control law determined from this weighting attempts
to drive the angular position and structure displacement to zero. The weighting matrix R is
chosen to have two different values to generate a control law with vibration suppression both
with and without the use of the piezoceramic actuator. For the case with the added piezo
actuator, the matrix R is chosen to be
Rl=diag[1.0 lx10 -4] (13)
G67
This choice penalizes the use of the motor voltage in factor of the piezoelectric actuator
voltage. For the case without piezo actuator control, the weighting matrix R is chosen to be
R2=diag[1.0 lx10 8] (14)
A comparison of the results of using the piezoceramic actuator versus using only the
motor torque for vibration suppression of the beam is illustrated in figures 5-8. In each case
the control or response without the advantage of the piezoceramic is given by the dashed line
and those with the use of the active beam are given by the solid lines. Figure 5 illustrates
that the voltage supplied to the armature of the motor is reduced by 33% (from - 3 volts to -2
volts) when slewing is performed on an active beam versus a passive beam. Figure 7 clearly
illustrates that the maximum tip deflection (overshoot) is reduced by almost 50% by using the
piezoceramic actuator.
4. Closing Remarks
This paper examines slewing control by introducing the concept of using an active
structure to improve performance. Slewing an active structure, as opposed to slewing a
passive structure, offers the advantage of reducing the peak voltage demands on the slewing
motor hence increasing reliability and potentially saving weight (a smaller motor could be
used). In addition the active structure approach promises to substantially reduce maximum
tip deflection of the structure. Simulation results were presented for a beam. These results,
although simulated, use experimentally measured parameters from laboratory tests of the
beam, motor and piezoceramics. The passive slewing beam model has been experimentally
verified using a PID control (Garcia, 1989).
The frame experiment of figure 4 is in progress. The finite element model is developed
and experimentally tested. The active strut has been designed and constructed and is being
installed in the frame. The key experimental results of interest are the strong coupling
between the bending vibration of the frame and the torsional vibration of the frame. While
this is to be expected, the flexibility of the frame (c01 = 1.6 Hz) enhances the problem of
suppressing tip vibration. Preliminary controllability calculations indicate that, unlike the
beam, the secondary piezoelectric actuator is needed to produce large enough control effort to
suppress the torsional modes.
In conclusion, the slewing of flexible structures requires a detailed examination of
control structure interaction and can benefit from the use of "smart" structures. The result is
best illustrated by examining the control input matrix Bf of equation (5). Without modeling
the interaction and flexibility of the structure the matrix, Bf is a scalar (i.e., r = 0). When the
interaction is modeled (r _: 0) the matrix Bf becomes a row vector. When the secondary
piezoceramic actuators are added, the matrix Bf becomes 2xn and the system is approaching
full state feedback which is known to yield the best performance.
0Fig. 4. The experimental slewing frame with active, piezoceramic struts in bending.
_u-mature voltage
-2
-3
0 0.1 0.2 0.3 0.4 0.5
riffle,Scc
Fig. 5. Voltage applied to the armature of the motor for each
of the two control laws.
0-o.2
-O.4
g -0.6
-0.8
-1
0
angular position
2 4 6
time, sec
8
Fig. 6. Angular position versus time for each of the two control laws.
0
-0.2
0.2 - structural tip deflection
F r _
///'
I l
-0.4
0 2 4 6 8
time, sec
Fig. 7. The deflection of the tip of the beam versus time for each
of the two control laws.
o
>
50
0
-50
-1001
-150 !
-200
Fig. 8.
References
voltage across.piezo actuator
-.--,-..,__.__
0 2 4 6 8
time, sec
The voltage applied to the piezoceramic versus time for each of the two control laws.
Burke S. B., and Hubbard J. E., 1987, "Active Vibration control of a Simply Supported
Beam Using a Spatially Distributed Actuator" IEEE Control Systems Magazine, Vol. 7 No. 4.
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Crawley, E.F. and Anderson, E.H.,1990, "Detailed Models of Piezoceramic Actuation
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