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Appendix A

' -- • . *-

THE LDCM ACTUATOR FOR VIBRATION SUPPRESSION *

Eric N. Ide and Douglas K. Lindner

Bradley Department of Electrical Engineering

Virginia Polytechnic Institute

Blacksburg, Va. 24061

ABSTRACT

A linear DC motor (LDCM) has been proposed as

an actuator for the COFS I Mast and the COFS program

ground test Mini-Mast. The basic principals of operation of the LDCM as an actuator for vibration

suppression in large flexible structures are reviewed. Because of force and stroke limitations,

control loops are required to stabilize the actuator,

which results in a non-standard actuator-plant configuration. A simulation model that includes LDCM

actuator control loops and a finite element model

of the Mast is described, with simulation results

showing the excitation capability of the actuator.

1.0 INTRODUCTION

A linear DC motor (LDCM) has .been proposed as

an actuator on the COFS-1 Mast ancf the COFS ground

test article Mini-Mast. This actuator has physical

limitations which impose limits on its ability to

generate force. The configuration of the

mast/actuator system gives rise to a non-standard

plant/actuator configuration. Feeding back the

digital relative position measurement causes impure

force generation. This paper will focus on how these

facts arise and how they affect the use of the LDCM

actuator for vibration suppression in large flexible

structures.

The paper will start with a description of the

LDCM actuator in Section 2. This section will include a description of the force limitations of the

actuator. Section 3 will describe the mast/actuator

system including compensation of the actuator.

Section 4 describes the simulations performed and

presents results.

2.0 ACTUATOR DESCRIPTION

The actuator proposed for use in the COFS program is a linear DC motor (LDCM). The LDCM consists

of two parts. The stationary base will mount on the

mast. It contains the motor coils and drive circu-

* This work was supported in part under NASA Grant

NAG-1-719.

itry. The reaction mass is the bar which moves on

bearings through the base. The bar contains permanent magnets to generate the motor flux.

A force applied to the reaction mass via the

motor coils will apply an equal and opposite force

to the base and hence, the mast. Two sizes of

actuators are being built. A Type I actuator has

an 11.6 kilogram reaction mass and a 15 centimeter

stroke. A Type II actuator has a 6.8 kilogram reaction mass and a 7 centimeter stroke.

The physical characteristics of the LDCM

actuator place limits on the amount of force it can

generate. The force generated by the reaction mass

Is governed by Newton's Law,

aa , (1)

where a is the second derivative of the reaction mass

inertial position. Looking at sinusoidal excitation

and taking the maximum gives,

-

 ttd<°

(2)

where d is the stroke length of the excitation.

Thus, at low frequencies, the maximum force available

is limited by the stroke Holt of the reaction mass.

At high frequencies, the maximum force is limited

by the power available to be delivered by the power

electronics. Fig. 1 shows the maximum force output

for the two types of actuators versus frequency.

3.0 SYSTEM DESCRIPTION

A model of the actuator mounted on the COFS Mast

is formed to examine system performance. Fig. 2

shows a representation of the actuator mounted on

the mast. The equations of motion of the reaction

mass are:

* =

F

U)CM (3)

where x = reaction mass inertial position. A finite

element model of the mast gives the mast equations

of motion as:

q + Dq +Kq =

P = Cq

(A)

echnic

CSCI 09C

8-239 (»(f

G3/33 01465T5
where P = mast inertial position. The force on the

reaction mass is equal end opposite to the force on

the mast.

(5)

''LDCM

 is directly related to the current input to

the actuator. These equations of motion are represented in Fig. 3, a block diagram of the uncontrolled LDCM actuator and mast. In this

configuration, the actuator is unstable. Hence,

actuator stabilization loops are required.

Fig. 4 shows a possible concept for stabilization of the actuator. The configuration is essentially position-velocity feedback. The velocity of

the reaction mass is obtained by integrating the

output of an accelerometer mounted on the reaction

mass. The only position measurement available is a

digital measurement of the reaction mass relative

position, the difference between the reaction mass

inertial position and the mast inertial position.

Feeding back the relative position places the mast

in the feedback path. Thus, the actuator control

loops couple the actuator to the mast in a feedback

configuration rather than the usual cascade configuration.

Also, feeding back the digital relative position measurement has other effects. Each time the

measurement device steps in quantization level, a

spike will be generated in the LDCM force. These

force spikes result in impure force being applied

to the structure.

4.0 SIMULATIONS

Simulations were used to evaluate the performance of the system. We will examine the behavior

of the COFS Mast with ten LDCM actuators mounted on

it. Four Type I actuators will be located at the

tip of the mast in a square configuration. This

allows the tip actuators to apply force in either

direction as well as torsionally. Six Type II

actuators will be placed in orthogonal pairs along

the length of the structure.

A finite element model was used for the mast.

Sixteen modes, 6 rigid body and 10 flexible, were

used in the model. All frequencies were below 7 Hz.

A model for each actuator was generated from

the block diagram in Fig. 4. The model contains all

nonlinearities such as force saturation, stroke

limits, and digital position measurement for all ten

actuators. A fourth-fifth order, variable step size

Runge-Kutta differential equation solver was used

in the simulations.

The system was excited by applying a sinusoidal

relative position command to one actuator. The

following figures show simulation results from an

excitation frequency corresponding to the second

bending mode in the x direction (f = 1.36 Hz). The

actuator used for excitation was the one which would

be most effective in exciting the mode being analyzed. All other actuators were given a zero command. After five cycles of excitation, all actuators

were given a zero relative position command to allow

the structure to free decay.

Fig. 5 shows the mast inertial position at the

actuator location where excitation occurred. The

mast model included a small amount of inherent

damping. The simulations showed an increase in

damping. Thus, the actuator stabilization loops

changed the behavior of the plant due to its location

in the feedback path.

Fig. 6 shows the force output of the same

actuator. Notice that the actuator continues to

generate force after the command signal has been

turned off. This is due to the component of the mast

position in the force.

Fig. 7 shows the power spectral density of the

LOCM force output of the exciting actuator during

the excitation phase. The force spiking due to the

digital relative position measurement generates the

noise level shown on the graph. This is an example

of the impure force generated by the actuator.

5.0 CONCLUSIONS

This paper has outlined the use of the LDCM

actuator for vibration damping on flexible structures. The stroke length and power electronics limit

the maximum force generated by the actuator. The

actuator must be stabilized because it is open loop

unstable. The available measurements for feedback

place the plant in the feedback path. Thus, the

actuator stabilization loops affect the plant behavior. Selected simulations showed the excitation

capability of the actuator and impure force characteristics.

FORCE

LIMIT

(N)

TIP ACTUATOR

SECONDARY ACTUATOI

O.I 1.0

FREQUENCY (Hz)

Fig. 1. Maximum Force Output
•ACTMIMSS

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Fig. 2. Actuator Mounted on Mast

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X RlACTION MASS SNOTtM. *O*ZTIQH

r NMT mnrrxAL MMITZON

6 POSITION COMMMD INMIT

F1

J

,_ 1

j

^

1

1

H

-1

P_

Q(s>

Fig. 4. Actuator Stabilization Loops

Fig. 3. Uncontrolled Actuator

Petition ....

TtaM(MC)

Fig. 5. Mast Inertial Position

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Fora ,

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Fig. 6. Force Output of Exciting Actuator

PSD

(»/•)

Fig. 7. Power Spectral Density of Force During Excitation