Modeling and Active Vibration Control of a Smart Structure

Nader Ghareeb and R¨udiger Schmidt

Institute of General Mechanics, RWTH Aachen University of Technology, Templergraben 64, Aachen, Germany

Keywords:

Super Element Model, Lyapunov Function Controller, Strain Rate Feedback Controller.

Abstract:

Active vibration control of ﬂexible structures has gained much attention in the last decades. The major com-

ponents of any active vibration control system are the mechanical structure suseptible to vibration, sensor(s) to

perceive it, actuator(s) to counteract the inﬂuence of disturbances causing vibration and ﬁnally, the controller

responsible for the generation of appropriate control signals. In this paper, a Lyapunov function controller and

a strain rate feedback controller (SRF) are used to attentuate the vibrations of a cantilivered smart beam excited

by its ﬁrst eigenmode. A super element (SE) with a ﬁnite number of degrees of freedom (DOF) is derived from

the modiﬁed and damped ﬁnite element (FE) model. The state-space (SS) equations of the same model are

also extracted. Controllers are applied to the SE and SS models and results are presented and compared.

1 INTRODUCTION

In modern engineering, weight optimization has al-

ways the highest priority during the design of struc-

tures. Despite all its advantages, it results in lower

stiffness and less internal damping which cause the

structure to become more sensitive to vibrations

which could even lead to its failure (Ghareeb and

Radovcic, 2009). One way to overcome this prob-

lem is to implement active or smart materials that

can be controlled in accordance to the disturbances

or oscillations sensed by the structure itself. The cou-

pled electromechanical properties of smart materials,

which are used here in the form of piezoelectric ce-

ramics, make them well suited for the use as dis-

tributed sensors and actuators for controling structural

response. In the sensor application, mechanically in-

duced deformations can be determined from measure-

ment of the induced electrical potential, whereas in

actuator applications deformation or strains can be

controled through the introduction of an appropri-

ate electric potential (Narayanan and Balamurugan,

2003). Active vibration control has been applied at

the beginning on ships (Mallock, 1905), and later

on aircrafts (Hort, 1934). The rapid developments

in this ﬁeld lead after that to the use of point actu-

ators and sensors to control ﬂexible systems based

on the knowledge of elastic mode frequencies and

mode shapes at their location (Balas, 1978). The

use of piezoelectric materials as actuators and sen-

sors for noise and vibration control has been illus-

trated extensively over the past 30 years. An ac-

tive vibration damper for a cantilever beam using a

distributed parameter actuator was designed by (Bai-

ley, 1984). Three different control algorithms to con-

trol the cantilevered beam vibration with piezoactua-

tors were developed and implemented by (Bailey and

Hubbard, 1985). A rigorousstudy on the stress-strain-

voltage behavior of piezoelectric elements bonded to

beams was presented by (Crawley and de Luis, 1987)

and (Crawley and Anderson, 1990). Many other re-

searchers have also investigated the implementation

and use of piezoelectric actuators like (Fanson and

Chen, 1986), (Preumont, 2002), etc.

The present work comprises modeling of a smart

beam and the design of active linear controllers to

control its vibrations. The piezoactuator is initially

modeled and the relation between voltage and mo-

ments at its ends is examined. A modiﬁed FE model

of the smart beam is then created. The damping co-

efﬁcients are calculated and added to it prior to the

reduction to a SE with a ﬁnite number of DOF. The

SS model is also extracted to check the controllers.

The FE- and SE models are validated by performing a

modal analysis and comparing the results to the exper-

imental ones. Finally, a Lyapunov function controller

and a SRF controller are introduced and implemented

on the SE and the SS models of the smart beam and

results are shown. The FE package SAMCEF is used

for the creation of the FE and the SE models, and

MATLAB-SIMULINK is used for the implementa-

tion of the controllers in the SS form.

142

Ghareeb N. and Schmidt R..

Modeling and Active Vibration Control of a Smart Structure.

DOI: 10.5220/0004004301420147

In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 142-147

ISBN: 978-989-8565-21-1

Copyright

c

2012 SCITEPRESS (Science and Technology Publications, Lda.)

2 MODELING

The ﬁrst step in designing a control system is to build

a mathematical model of the system and disturbances

causing the unwanted vibrations. The structural ana-

lytical model can be derived by using the FE method.

The smart structure used here consists of a steel beam,

a bonding layer and an actuator (Figure 1).

V

beam

actuator

bonding layer

Figure 1: The smart beam.

2.1 Actuator Modeling

The introduction of an actuator implies the imple-

mentation of an appropriate electric potential to

control the vibrations in the smart structure (converse

piezoelectric effect). The voltage applied can be

represented by two equal moments with opposite

directions concentrated at it’s ends (Fanson and Chen,

1986). The behavior of the piezoelectric material is

assumed to be linear thoughout this work. By consid-

ering the schematic layout of the middle portion of

the smart beam (Figure 2), if a voltage V is applied

across the piezoelectric actuator while assuming one

dimensional deformation, the piezo-electric strain ε

p

in the piezo is:

M

p

y

b

D

t

3

t

2

t

1

z

beam

actuator

bonding

x

z

M

p

Figure 2: A schematic layout of the composite beam.

ε

p

=

d

31

t

1

·V (1)

V is the voltage of the piezo-electric actuator, d

31

the

electric charge constant and t

1

it’s thickness.

The longitudinalstress can be expressed with Hooke’s

law as:

σ

p

= E

1

· ε

p

(2)

where E

1

is the Young’s modulus of the piezo

This stress generates a bending moment M

p

around

the neutral axis of the composite beam given by:

M

p

=

Z

(t

1

+t

2

+t

3

−D)

(t

2

+t

3

−D)

σ

p

· b· zdz (3)

By considering the equilibrium of moments about the

neutral axis:

Z

beam

σ

3

dA +

Z

adhesive

σ

2

dA +

Z

piezo

σ

1

dA = 0 (4)

the position of the neutral axis D is found.

Substituting D, (1) and (2) in (3) determines the bend-

ing moment generated by the piezo M

p

as a function

of the voltage V:

M

p

=

E

1

E

2

(t

1

t

2

+ t

2

2

) + E

1

E

3

(t

2

3

+ t

1

t

3

+ 2t

2

t

3

)

2(E

1

t

1

+ E

2

t

2

+ E

3

t

3

)

d

31

bV

Now, the actuator moment can be taken instead of

the voltage as input to the controllers that will be de-

signed and implemented.

2.2 FE Modeling of the Smart Beam

The resultant FE model of the smart beaam must be

faithfully representative so that it can be used for fur-

ther applications like control analaysis (He and Fu,

2001). In order to ﬁnd the best FE model, the opti-

mal element type and size must be selected. Thus,

a modal analysis of the real beam is experimentally

performed and then the results are compared to those

from the FE with different element types used. A de-

tailed geometry of the smart beam is shown (Figure 3)

and the material properties and thickness of each part

are represented (Table 1).

layer

1307510

5

10

10

5

composite

Figure 3: A detailed geometry of the smart beam [dimen-

sions in mm].

Table 1: Parameters of the components of smart beam.

Beam Bonding Actuator

Material steel epoxy PIC 151

Thickness [mm] 0.5 0.036 0.25

Density [kg/m

3

] 7900 1180 7800

Young’s mod. [GPa] 210 3.546 66.667

ModelingandActiveVibrationControlofaSmartStructure

143

Table 2: Comparison between FE and SE models.

FE model Experiment

1

st

eigenfrequency [Hz] 13.81 13.26

2

nd

eigenfrequency [Hz] 42.67 41.14

2.2.1 FE - Type and Size

The smart beam is modeled as a composite shell with

three layers, but without any relative slip among the

contact surfaces. Hence, each layer has its own me-

chanical properties. To validate the element type

used, a modal analysis is done and the ﬁrst two eigen-

frequencies are read and compared to those from the

experiment (Table 2).

Although reducing the element size can improve

the solution accuracy, but the use of excessively ﬁne

elements may result in unmanageable computations

that exceed the memory capabilities of existing com-

puters (Ko and Olona, 1987). The analysis shows that

using an element size less than 1 mm doesn’t make

any signiﬁcant change on the values of the 1

st

and 2

nd

eigenfrequencies of the smart beam.

2.2.2 Damping Characteristics

Damping parameters are of signiﬁcant importance in

determining the dynmaic response of structures (Lee

and Davidson, 2004). In this work, the damping is as-

sumed to be viscous and frequency dependent for the

sake of convenience and simplicity. A special case

of viscous damping is known as the proportional or

Rayleigh damping. The damping matrix is thus ex-

pressed as a linear combination of mass and stiffness

matrices of the undamped model (Rayleigh, 1877):

C = αM + βK (5)

where α and β are real scalars that need to be de-

termined. To do that, many methods are available

(Spears and Jensen, 2009) and (Adhikari, 2000).

However, a method that was developed by (Chowd-

hury and Dasgupta, 2003) is used in this work.

3 THE SUPER ELEMENT

The main virtue of this technique is the ability to per-

form the analysis of a complete structure by using re-

sults of prior analysis of different regions comprising

the whole structure. Thus, all DOF considered useless

for the ﬁnal solution will be condensed and the rest

will be retained. This means, the DOF of the whole

system will correspond to the retained nodes plus a

number of internal deformation modes. To construct

the SE, the component-mode method is used (Craig

and Bampton, 1968), (Rickelt-Rolf, 2009).

3.1 The Component-mode Method

also called method of Craig-Bampton, it implies that

the basic substructure is split into a certain number of

substructures. The DOF of each substructure are then

classiﬁed into boundary DOF and internal DOF. The

boundary DOF are shared by several substructures,

while the internal DOF belong only to the considered

substructure. The behavior of each substructure is de-

scribed by the combinationof two types of component

modes; the constrained modes and the normal vibra-

tion modes. The former are determined by assigning

a unit displacement to each boundary DOF while all

other boundaries DOF are being ﬁxed. The latter cor-

respond to the vibration modes obtained by clamping

the structure at its boundary. It is thus assumed that

the behavior of the substructure in the global system

can be represented by superimposing the constrained

modes and a small number of normal modes. Hence,

by retaining only the low-frequency vibration modes,

the substructure’s dynamic deformed shape is repre-

sented with sufﬁcient accuracy.

3.2 SE Modeling

In order to construct a SE, the retained nodes and the

condensed nodes must be designated and the num-

ber of internal modes to be used must be speciﬁed.

The retained nodes are usually those where boundary

conditions are applied, or where stresses, displace-

ments,...etc are applied or measured. Based on the

current work, 10 internal modes responding to about

95% participation of the mass are used and ﬁve re-

tained nodes are considered (Figure 4). Node 1 is

clamped, actuator moments are added on node 2 and

node 3. Node 4 is used as a secondary sensor (for fu-

ture works), and node 5 is used as a sensor to measure

the tip displacement. A comparison between the FE

and the SE models is shown in (Table 3) and (Table 4).

Results of modal analyses on both models don’t show

a big difference concerning the eigenfrequencies.

1 54

32

Figure 4: The retained nodes of the super element.

4 THE STATE SPACE MODEL

The fundamental equation describing the dynamic

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

144

Table 3: Characteristics of FE and SE models.

FE model SE model

Number of nodes 8206 5

Number of elements 2575 1

DOF 49236 34

Table 4: Comparison between the eigenfrequencies.

FE model SE model % Error

1

st

freq [Hz] 13.811 14.249 3.07

2

nd

freq [Hz] 42.673 43.414 1.71

behavior of a damped structure discretized by the FE

method is written in the form:

M ¨q(t) + C ˙q(t) + Kq(t) = f(t) (6)

where q(t) is the state vector which collects the dis-

palcements of the structure by DOF, and the f(t) vec-

tor expresses the applied loads. If the total number of

DOF of the FE model is n, then the dimension of the

state vector q(t) is also n, and that of the mass, stiff-

ness and damping matrices will be n ∗ n. However,

since a SE is constructed and the desired results only

concern speciﬁc locations of the structure, n will be

reduced to s. In this case:

s = r + m − 6 (7)

r and m are the number of retained nodes, and the

number of internal modes chosen.

The SS model is written according to (6) with dimen-

sions of the SE model. The general form is:

˙x = Ax + Bu, and y = Cx + Du (8)

x is the state vector, u and y are the input and output

vectors. By using a Fortran code, and upon specifying

the type and position of inputs and outputs, the SS

model of the SE model is created.

5 CONTROLLER DESIGN

The performance of smart structures for active vibra-

tion control depends strongly on the control algorithm

accompanied with it. The objective is to design con-

trollers capable of damping the vibrations due to the

ﬁrst eigenmode of the smart beam. The beam is ini-

tially excited with its ﬁrst eigenmode and then it is

left to vibrate freely. At this moment, the controllers

are activated. Two vibration suppression methods are

used; Lyapunov stability-based theorem control and

the SRF control.

5.1 Lyapunov Stability Theorem

Although there is no general procedure for construct-

ing a Lyapunov function, yet any function can be con-

sidered as a candidate if it meets some requirements,

i.e. positive deﬁnite, equal to zero at the equilib-

rium state and with its derivative less or equal to zero

(Khalil, 1996). Now, the energy equation of a thin

Bernoulli-Euler beam modeled as a single FE in a

one-dimensional system with length h and left point

coordinate x

i

, will be considered as a Lyapunov func-

tion candidate:

U =

1

2

Z

x

i+h

x

i

ρ

∂y

∂t

2

+ EI

∂

2

y

∂x

2

2

dx (9)

This function is locally positive deﬁnite, continuously

differentiable and equal to zero at the equilibrium

state. Yet, the derivative of this function must be

smaller or less than zero.

˙

U =

Z

x

i+h

x

i

ρ˙y¨y + EI

∂

2

y

∂x

2

∂

∂t

∂

2

y

∂x

2

dx (10)

Substituting the equations for the bending moment M

and shear forceV in a beam (Petyt, 2003) and assum-

ing there is no shear yields:

˙

U = M

˙

y′

x

i

+h

−

˙

y′

x

i

(11)

To ensure that (11) is always smaller or equal to zero,

the actuator moment M can have the value:

M = − k

˙

y′

x

i

+h

−

˙

y′

x

i

(12)

where k is a positive constant. Substituting (12) in

(11) yields:

˙

U = − k

˙

y′

x

i

+h

−

˙

y′

x

i

2

≤ 0 (13)

In this case, (12) can be used as the controller for the

smart beam.

5.1.1 SE Model with Controller

Based on the SE model created, two equal moments

at nodes 2 and 3 (but with opposite directions) are

added, where each one of them is a function of the

velocities at both nodes according to (12). The effec-

tiveness of this control strategy is shown in (Figure 5),

and its effect on the magnitude of the eigenmode on

the FFT spectrum diagram is illustrated in (Figure 6).

5.1.2 SS Model with Controller

In the SS, two steps are done. In step one, the only

input is the forced excitation with the ﬁrst eigenmode

ModelingandActiveVibrationControlofaSmartStructure

145

20 22 24 26 28 30 32 34 36 38 40

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time (s)

Tip displacement (m)

No control

Lyapunov stability theorem

Figure 5: Tip displacement vs. time with and without con-

troller during the free vibration (SE model is used).

0 5 10 15 20 25

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency (Hz)

Magnitude (mm)

No control

Lyapunov stability theorem

Figure 6: The FFT spectrum of the smart beam.

v

3

y = Cx + Du

x′ = Ax + Bu

Tip displacement

0

k

M

3

M

2

v

2

Figure 7: The SS model of the smart beam with controller.

and the output consists of the state vectors that are

fed as initial conditions for the second step. In step

two, (Figure 7), the input consists of both actuator

moments. The output comprises the tip displacement

at the 5

th

retained node, and the velocities at the 2

nd

node and 3

rd

node. Based on the Lyapunov stability

theorem, the output multiplied by a constant k is fed

back as actuator moments with different signs.

5.1.3 Comparison of Results from Both Models

The controller is implemented on both models and a

very slight difference is seen if the output curves are

zoomed (Figure 8). This is due to the fact that differ-

ent time steps are used in both models. Thus, using a

SE model decreases the simulation time and can show

more results (e.g. stresses, energy curves, etc).

5.2 Strain Rate Feedback Control

The SRF control can be used for active damping of

a ﬂexible space structure (Fei and Fang, 2006). The

20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 21

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time (s)

Tip displacement (m)

SS control

SE control

Figure 8: Tip displacement vs. time with both models.

structural velocity coordinate is fed back to the com-

pensator, and the compensator position coordinate

multiplied by a negative gain is fed back to the struc-

ture. The SRF model can be presented with the fol-

lowing equations:

¨

ξ + 2ζ

˙

ξ + ω

2

ξ = − Gω

2

η (14)

¨

η + 2ζ

c

ω

c

˙

η + ω

c

η = ωc

2

˙

ξ (15)

ξ is the modal coordinate of structure displacement,

ζ is the damping ratio of the structure, ω is the

it’s natural frequency, G is the feedback gain, η

is the compensator coordinate, ζ

c

is the damping

ratio of the compensator and ω

c

it’s natural frequency.

Since all the parts of the smart beam are inter-

grated in a single SE, it is supposed that the smart

beam and the controller have the same damping

ratio and the same eigenfrequency. Applying this

assumption in the equation of the smart beam gives:

¨

ξ+ 2ζω

˙

ξ+ (ω

2

+ Gβω

2

)ξ = 0 (16)

In this case, there will be an increase in the stiffness

of the structure (active stiffness). However, the stabil-

ity condition is not clearly deﬁned here due to the fact

that the closed-loop damping and stiffness matrices

of the whole system cannot be symmetrized (New-

man, 1992). The effectiveness of the SRF controller

is shown in (Figure 9). The magnitude of the eigen-

mode of the system decreases as well (Figure 10).

6 CONCLUSIONS AND FUTURE

WORK

In this work, the basic procedures for the modeling

and simulation of a smart beam were presented and

the SE and SS models were derived. A Lyapunov

function and a SRF controllers were designed and im-

plemented, and their effectiveness was checked.

In the future, an observer will be designed to count

for the geometrical nonlinearities which were not con-

sidered in this work. Nevertheless, more eigenmodes

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146

20 22 24 26 28 30 32 34 36 38 40

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time, (s)

Tip displacement, (m)

No control

SRF control

Figure 9: Tip displacement vs. time with the SRF controller.

0 5 10 15 20 25

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency (Hz)

Magnitude (mm)

No control

SRF control

Figure 10: The FFT spectrum of the smart beam.

will be controlled, and other control strategies (e.g.

positive position feedback, PID controllers...etc) will

be investigated and implemented.

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