NITINOL-reinforced plates: Part III. Dynamic characteristics

Composites Engineering, Vol. 5, No. 1, pp. 91-106, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0961-9526/95 $9.50 + .00

Pergamon

0961-9526(94) E00086-7

NITINOL-REINFORCED P L A T E S : P A R T III. DYNAMIC CHARACTERISTICS t J. Ro and A. Baz Mechanical Engineering Department, The Catholic University of America, Washington, DC 20064, U.S.A. (Received 8 July 1994; final version accepted 27 September 1994) Abstract--The dynamic characteristics of NITINOL-reinforced composite plates are controlled by heating sets of NITINOL fibers embedded inside these plates. The activation of the shape memory effect of these NITINOL fibers increases the elastic energy, enchances the stiffness of the composite plates and modifies their modal characteristics. One of the objectives of the resulting modal modification is to shift the modes of vibration of the plates away from the excitation frequencies in order to avoid undesirable resonances. In this way, the modal characteristics can be tailored in response to the external disturbances acting on the plates. The classical finite element approach is used to form the equations of motion of the assembly of NITINOL-reinforced plate elements and the appropriate boundary conditions are then applied. The solution of the eigenvalues of the resulting homogeneous equations gives the natural frequencies of the NITINOLreinforced plate as influenced by the properties of the composite matrix and the NITINOL fibers. It is important to note that these properties are influenced by the temperature distribution inside the composite plate which is developed by virtue of activating and de-activating the NITINOL fibers. Emphasis is placed on the effect of intentional electrical heating of a selected subset of the NITINOL fibers on the overall dynamics of the plates. The effect of the associated thermal energy propagating through the composite on the unintentional thermal activation of additional subsets of the NITINOL fibers is accounted for. Such an effect is not only significant, but also essential to the thorough understanding of the operation of the NITINOL-reinforced plates. NOMENCLATURE A a

b E

[fA h

[gel [KB,I [KGe] KE

[MA Nx,y, xy

t U W

w, w2 we x, y P V

[~'] [w] Ix]

surface area of the unit element the length of the rectangular element the width of the rectangular element Young's modulus of the material nodal forces vector thickness of the plate elements stiffness matrix flexural stiffness matrix geometric stiffness matrix kinetic energy consistent mass matrix in-plane force in x, y and xy direction per unit length time strain energy transverse deflection work done due to the nodal forces work done due to the in-plane forces total work done Cartesian coordinates along the plate neutral axis, respectively mass per unit volume Poisson's ratio of isotropic material 12 × 1 nodal displacement vector of the four nodes of the plate element 12 x 12 matrix depending on nodal coordinates curvature 1. INTRODUCTION

Shape memory alloys, such as the Nickel-Titanium alloy (NITINOL), have been utilized extensively in the development of a wide spectrum of smart composites (Ikegami et al., 1990; Roger et ai., 1991 and Baz et al., 1990-1994). Such wide acceptance of these alloys stems from their unique properties and behavior when subjected to particular heating and cooling strategies. For example, large changes in Young's modulus are encountered when these alloys undergo their unique martensitic phase transformation (Funakubo, 1987). t This paper was submitted for presentation at the International Conference on Composites Engineering (ICCE/1), New Orleans, 28-31 August 1994. 91

92

J. Ro and A. Baz

Also, these alloys are capable of producing remarkable recovery forces and/or inelastic strains when heated above their phase transformation temperature. These properties rendered the shape memory alloys suitable for use as control actuators which can transform thermal energy directly into mechanical work (Perkins, 1975; Duerig et al., 1990). Accordingly, if NITINOL fibers are embedded inside a composite matrix at optimal locations, they can be used to control the static and dynamic characteristics of the resulting smart composite. The control action is generated either by the stiffening of the NITINOL fibers and/or the shape memory effect. With such built-in control capabilities, the performance of the smart composites can be optimized and tailored to match changes in the operating conditions. It is therefore the purpose of this study to develop a thorough understanding of the fundamentals governing the performance of the NITINOL-reinforced plates. The individual contributions of the composite matrix, the NITINOL fibers, the shape memory effect and thermal distribution inside the composite to the overall dynamic behavior of the NITINOL-reinforced plates will be determined. This paper is organized in four sections. In Section 1, a brief introduction is given. In Section 2, the mechanical finite element model describing the dynamics of the NITINOL-reinforced plate is developed. The experimental dynamic behavior of the NITINOL-reinforced plate is given in Section 3. Comparison between the theoretical and experimental frequencies and mode shapes are also presented for different activation strategies. Section 4 summarizes the conclusions of the present study. 2. FINITE ELEMENTMODELINGOF NITINOL--REINFORCEDPLATES In the present study, the NITINOL-reinforced composite plates are made by embedding the NITINOL fibers inside vulcanized rubber sleeves placed along the neutral planes of these composite plates as shown in Fig. 1. In this arrangement, the NITINOL fibers are free to move during the phase transformation process in order to avoid degradation and/or destruction of the shape memory effect which may result when the fibers are completely bonded inside the composite matrix. The dynamic characteristics of this class of NITINOL-reinforced plates are obtained by applying the mechanical finite element model developed in the following section along with the thermal finite element model presented in Part I of this paper. The overall stiffness and mass matrices of the entire NITINOL-reinforced plate are obtained by assembling the stiffness and mass matrices of the individual elements. 2.1. Theoretical formulation 2.1.1. Strain energy. According to Classical Plate Theory, the normal stress trz, and shear strains Yyz and y= are negligible. Thus, the overall strain energy U may be written as (Young, 1986):

u=24(i-v~)

If

A \~X 2) + \OY ~j + 2(1 -- V)[0---k-~Y) + V~x2 ~flJ dxdy' (1)

where E, v, A and h denote modulus of elasticity, Poisson's ratio, area and thickness of the plate. Also, w defines the transverse deflection of the plate at any location (x, y). 2.1.2. External work done. The external work done We on the unit NITINOL-plate element consists of two parts. The first part ~ is due to the nodal forces [fn]. The second part We is due to the in-plane forces which include the recovery forces of the NITINOL fibers, thermal load induced by heating the NITINOL fibers and the external in-plane loads. The work ~ is given by: W~ = ½[f~lr[6e] = l[ge]r[Ke][c5~ ]

(2)

NITINOL-reinforced plates: Part III. Dynamic characteristics

93

b

//~,, x

/

SLEEVE -/

f

T

)4 COMPOSITES

/

"~.- NITINOL FIBER

Fig. 1. A schematic drawing of a NITINOL-reinforced plate.

where [Ke] is the element stiffness matrix and [die] is the element displacement vector. The work done W2 by the in-plane loads (N x, ivy, and Nxy) is given by:

h Sf

[

t/Ow\2

/3w\ 2

3(~x)(Ow)]

(3)

where

Nx=Pmx+m,x-Fwx, Ny = Pray + Pry,

(4)

Nxy = Pm~ + Pe~, with Pmx,y.~ and Ptx.y.~ denoting the compressive in-plane mechanical and thermal loads in the x, 3' and xy directions. Also, Fwx denotes the total tension developed in the N I T I N O L fibers which are aligned along the x-axis. Such a tension is due to initial tightening and phase recovery forces resulting from activating the shape memory effect. 2.2. Finite element displacement formulation The spatial distribution of the transverse deflection w, over a unit element of the NITINOL-reinforced plate can be expressed by the following 12-term polynomial:

w = c~ + a2x + ot3y

+ O/4x2 q'- Ol5XY + ot6Y2 + or7x3 +

+ ~ 9 x Y 2 + OQoy 3 + ~ l l X 3 y

+ o~12xY 3.

asxZY (5)

If [di] is the vector of the nodal displacements and [a] is the vector of the coefficients ai, i = 1, 2 . . . . . 12, then the nodal displacement can be written as [di] =

w

Ox

= [Fl[od

(6)

where [F] is a known 3 × 12 matrix. The unknown constants oti can be calculated in terms of the nodal displacements if the nodal coordinates are successively introduced into eqn (6). The resulting matrix equation may be solved for the constants ai yielding

[~] = [w]-~[die]

(7)

where [W] is 12 × 12 matrix whose entries depend on nodal coordinates, [die] is the 12 x 1 nodal displacement vector of the four nodes. Based on eqns (6) and (7), the curvature [x]

J. R o a n d A . Baz

94

and angular displacement [0] vectors are written as follows (Chajes, 1974): [x] = [0] =

_O2w Ox2

_32w _202w]r Oy2

Oxay J = [Cl[od = [BI[J~I,

low o_w] Ox

igyJ

= [Gl[a] = [H][Jel,

(8) (9)

where [C] is a 3 x 12 matrix, [B] = [C]pF] -1, and [G] is a 2 × 12 matrix and [H] = [G]pF] -~. 2.3 Stiffness matrix Equilibrium of the element is attained when the external work done becomes equal to the strain energy, We = WI + We. Thus from eqns (1)-(3), and eqns (8)-(9), we have:

[Blr[DI[B]d x d y [ J e]

[J~lr[KA[Jq = [J~l r o

o

[HIr[NI[HI d x d y [ J q ,

- [J~l r J0

(10)

0

i.e.

(11)

[gel = [Kne] - [Keel where

[Kn,] =

[B]T[DI[B]dx dy o

and

[K~.] =

fof b[Hlr[N][HI dxdy o

o

(12)

o

where [D] is flexural rigidity matrix and IN] = [ Nx In eqn (11), [Kae] denotes the conventional flexural stiffness matrix associated with the bending vibration of the thin rectangular plate element. For a NITINOL-reinforced plate, the dependence of the modulus of the elasticity of the matrix and flexural rigidity matrix [/9] on the temperature is particularly emphasized. Also, [K6~] defines the geometric stiffness matrix which results from the in-plane loads. Based on eqn (11), it is easy to understand the role that the NITINOL fibers play in controlling the overall stiffness matrix. For example, heating the NITINOL fibers in a clamped plate reduces the modulus of the elasticity of the matrix and generates in-plane compressive thermal loads. These two effects tend to reduce the overall stiffness of the plate. If the heating is high enough, such that the determinant of the overall stiffness becomes equal to zero, the plate will buckle. However, it is important to note that the tensile phase recovery forces which are developed by NITINOL shape memory effect, tend to increase the overall stiffness matrix to resist the plate buckling. Therefore, the NITINOL fibers can be used to compensate for the softening of the composite matrix and counterbalance the effect of the compressive thermal loads. Accordingly, the NITINOL fibers can be utilized to alter the structural frequency and increase the critical buckling loads of the plate in response to the external loads acting on it. 2.4. Finite element displacement formulation of consistent mass matrix The kinetic energy KE of the thin plate element can be calculated from:

f :°we where p is mass per unit volume. In eqn (13), the longitudinal and rotary inertia are neglected according to the Classical Plate Theory. According to the assumed shape

NITINOL-reinforced plates: Part III. Dynamic characteristics

95

function, the transverse deflection w can be expressed in terms of the nodal displacements joel as follows: w = [P][a] = [p][~pl-t[t~e] = [R][t~], (14) where [P] is a known 1 × 12 matrix. Equation (13) can be used along with eqn (14) to determine the consistent mass matrix of the element [Me] as follows: ½ [ ~ l r [ M e ] [ ~ l = 2J-[~lr

faI' o

such that

[Me]

[Rlrph[Rl d x d y [ ~ ] ,

(15)

o

[Rlrph[R] dxdy.

= o

(16)

o

2.5. Equation o f motion o f the NITINOL-reinforced plate The equation of motion of the NITINOL-reinforced plate element can be written as: [Me][~ e] + [/(Be - Kcel[J e] = 0,

e = 1. . . . . n.

(17)

The in-plane forces developed by the N I T I N O L fibers are accounted for by the geometric matrix which plays an important role in controlling the static and dynamic characteristics of the NITINOL-reinforced plates. By properly adjusting the pre-strain level of the N I T I N O L fibers, the recovery force can be controlled in such a way that the natural frequencies of the plate can be increased and the critical buckling loads can be enhanced in response to the external excitations acting on the plate. 2.6. Numerical results In this section, the eigenvalue problem corresponding to the assembled equations of motion of the elements is solved. The results obtained for un-reinforced plates are compared with the results available in the literature in order to validate the accuracy of the present finite element model. The model is then utilized to predict the dynamic behavior of NITINOL-reinforced plates. Table 1 lists the frequency parameter f~ for an isotropic un-reinforced square plate, with side a - - 0 . 1 9 0 5 m and thickness h = 0.00254m, made of fiberglass/resin such that E l = 4 . 0 G P a , p = 1499.0kg/m 3, and v = 0.33 with ~ = t O n a 2 ~ o ) and D O = E~h~/[12(1 - v2)]. The table shows that the present finite element model agrees extremely well with those of Gorman (1976) and Narita et al. (1992). In present finite element method, the plate is divided into 49 elements. Table 1. The frequency parameter f~, for isotropic un-reinforced square plate Mode

1

2

3

4

5

G o r m a n (1976) Narita and Leissa (1992) Present FEM

3.459 3.459 3.459

8.356 8.358 8.360

21.09 21.09 21.28

27.06 27.07 26.94

30.55 30.56 30.58

3. E X P E R I M E N T A L C H A R A C T E R I S T I C S OF N I T I N O L - R E I N F O R C E D P L A T E S

In this section, the experimental characteristics of a NITINOL-reinforced plate are presented. The effect of the activation strategy and initial tension of the NITINOL fibers on the natural frequencies and mode shapes are investigated experimentally. The vibration attenuation characteristics o f the plate are determined also when the plate is excited mechanically with random vibrations and the N I T I N O L fibers are controlled with various activation strategies. The experimental results are compared with the theoretical predictions.

J. Ro and A. Baz

96

~

ACCELEROMETER

' 64

/

/ S

ai\

GP-IB : FFT ANALYZER . . . . . ~ COMPUTER I

i T

S

~

i

'STAR MODAL PACKAGE

I~iP/CNITI~C~Ii i iNFORCED i COMPOSITE PLATE Fig. 2. Modal analysis set-up of the NITINOL-reinforcedplate. 3.1. Experimental performance of the NITINOL-reinforced plate

A NITINOL-reinforced plate is made of randomly oriented glass fibers and a low cure temperature polyester resin (Fibre Glass-Evercoat Co. Inc., Ohio 45242) matrix. The plate is 0.1905 m long, 0.1905 m wide and 2.54 mm thick. It is mounted with all its edges supported in a clamped arrangement. A total of 58 NITINOL fibers (Innovative Technology International Inc., MD 20705), that are 0.55 mm in diameter, are embedded in the mid-plane and along the x-direction inside the plate through vulcanized rubber sleeves that have outer diameter of 0.95 mm. The NITINOL fibers have a 50°C phase transformation temperature. The experimental set-up used for modal analysis tests is shown in Fig. 2 and the set-up used for monitoring the vibration attenuation characteristics is shown in Fig. 3(a) and (b). The purpose of the modal analysis tests is to determine experimentally the effect of the activation strategy and the initial tension of the NITINOL fibers on the natural frequencies and mode shapes of the plate. A modally-tuned impact hammer (model 086C80, PCB Piezotronics, Inc., NY), is used as a source of excitation. The response of the plate is monitored by a miniaccelerometer (model 309A, PCB). Signals from the impact hammer and accelerometer are sent to the FFT analyzer (CF-900, ONO SOKKI Co., MI) to determine the natural frequencies of the plate. The FFT analyzer is also connected to a micro-computer via a GPIB communication channel in order to identify the mode shapes using the STAR MODAL software package (V:2.0 GenRad Co., Milpitas, CA). The plate is divided into 49 equal elements along the x and y axis, by means of an 8 × 8 grid. The grid nodal points are marked as shown in Fig. 2 and the miniaccelerometer is placed at node 38 to monitor the response of the plate due to impacts at the remaining unclamped nodal points. The Frequency Response Function (FRF) data obtained are utilized to determine the natural frequencies and mode shapes. In this study, the energy used for activating the NITINOL fibers is 6.8 W/fiber. Also, the pre-tensions are set at either 0.0 or 31.14 N/fiber. 3.2. Modal characteristics Tables 2 and 3 list the natural frequencies of the NITINOL-reinforced plate, when the initial tension of the fibers are 0 and 31.14 N/fiber, respectively. The tables also show the effect of two activation strategies on the first four modes of vibration of the plate in comparison with those of a plate with unactivated fibers. The considered activation strategies energize either all or half the embedded NITINOL fibers. When the initial tension is equal to zero, Table 2 indicates that the first four modes of vibration of the plate with unactivated fibers are 217.52 Hz, 436.89 Hz, 444.84 Hz and 631.57 Hz, respectively. When all the fibers are activated the corresponding values become 201.65 Hz, 395.91 Hz, 441.63 Hz and 612.95 Hz. This corresponds to a decrease of 7.29%, 9.38070, 0.72°70 and 2.95070 relative to the modes of the plate with unactivated fibers. This decrease is attributed to the developed in-plane thermal loads and to the reduction of the modulus

NITINOL-reinforced plates: Part Ill. Dynamic characteristics

STINGER

ELECTROMAGNETIC SHAKER

•4 ~

POWER AMPLIFIER

~ ___

[

~

97

ACCELEROMETER

~gOhT~E~p~'~7~CED

NOISE GENERATOR

(a)

NITINOL

FIBERS

,v

FRAME

SHAKER

(b) Fig. 3. Set-up for monitoring the vibration attenuation characteristics of the NITINOLreinforced plate. (a) Schematic drawing, (b) photograph.

NITINOL-reinforced plates: Part lII. Dynamic characteristics

99

Table 2. The natural frequencies (Hz) measured with impact hammer for pre-tension of 0.0 N/fiber Mode

1

Unactivated Half fibers activated All fibers activated

217.52 208.54 201.65

2 436.89 413.69 395.91

3

4

444.84 463.97 441.63

631.57 621.64 612.95

Table 3. The natural frequencies (Hz) measured with impact hammer for pre-tension of 31.14 N/fiber Mode

1

2

Unactivated Half fibers activated All fibers activated

237.74 261.49 276.21

446.10 444.48 451.01

3 506.53 552.43 595.44

4 688.10 722.62 743.84

of elasticity of the matrix due to the heating of the NITINOL fibers. The fact that the NITINOL tension is initially zero makes the total tension, which is developed after the activation of the fibers, insufficient to counterbalance the effect of the thermal loads and softening of the matrix. When the pre-tension is increased to 31.14 N, the phase recovery forces become large enough to compensate for both these effects, as can be seen from Table 3. In the table, the first four modes of vibrations for the case of unactivated plate are 237.74 Hz, 446.10 Hz, 506.53 Hz and 688.10 Hz, whereas those with all fibers activated are 276.21 Hz, 451.01 Hz and 595.44 Hz and 743.84 Hz, respectively. This corresponds to an increase of 16.18%0, 1.1%0, 17.55% and 8.1% relative to the modes of the plate with unactivated fibers. The associated temperature distribution is shown in Fig 4(a). When half the fibers are activated, the temperature distribution is shown in Fig. 4(b). Because of the effect of unintentional thermal activation, the first four modes of vibration become higher than expected and assume values of 261.49 Hz, 444.48 Hz, 552.43 Hz and 722.62 Hz, respectively. Figure 5 shows a typical FRF for the different initial tensions and strategies. Figure 6 shows a plot of the mode shapes as obtained with the STAR MODAL software package for the different activation strategies when all the NITINOL fibers are pretensioned to 31.14 N/fiber. According to the figure, the mode shapes are insignificantly affected by the activation strategy as compared to the case of the plate with unactivated fibers. The reason is that the NITINOL/resin laminated plate has a single layer of NITINOL fibers which are placed at the mid-plane. Reviewing the results obtained, it is evident that the modification of the structural frequencies of the NITINOL-reinforced plate is clear and significant. Note that the achieved increase of the fundamental frequency of the plate with all fibers activated is 16.18% as compared to that of an unactivated plate while the volume fraction of the NITINOL fiber is only 2.85%, and the power used is only 6.8 W/fiber. The significant role that the NITINOL fibers play in tuning the modal parameters of the plate can contribute significantly to the effectiveness of the NITINOL-reinforced plates in attenuating structural vibrations as well as sound radiation.

3.3. Attenuation of structural vibration The experimental set-up used for monitoring the effectiveness of NITINOL reinforcements in attenuating the vibration of the plate is shown in Fig. 3. A random noise generator (Type 1405, B&K, Denmark) is connected to a power amplifier (model PA7C, Wilcoxon Research, MD 20814) to drive an electromagnetic shaker (Type F3, Wilcoxon Research, MD 20814). The shaker subjects the NITINOL-reinforced plate to broadband random vibration covering a frequency range up to 20 K. The plate response to these excitations is monitored by an accelerometer whose signal is sent to an FFT analyzer to determine the frequency content and the amplitude of vibration.

100

J. Ro and A. Baz

G

Fig. 4. The FEM predictions of the temperature distribution on the NITINOL-reinforced plate: when (a) all the NITINOL fibers are activated, and (b) half of the NITINOL fibers (along x axis) are activated. Tables 4 a n d 5 list the e x p e r i m e n t a l results for the first four modes o f v i b r a t i o n for two levels o f pre-tensions a n d for three activation strategies. Figure 7(a) a n d (b) shows the c o r r e s p o n d i n g f r e q u e n c y response for these different cases. T h e values o f the n a t u r a l frequencies o b t a i n e d are slightly higher t h a n those o b t a i n e d by the imapct h a m m e r m e t h o d due to the stiffness o f the excitation stinger. This effect makes the plate stiffer a n d the n a t u r a l frequencies higher. Table 4. The natural frequencies (Hz) measured with random excitation for pre-tension of 0.0 N/fiber Mode

1

2

3

4

Unactivated Half fibers activated All fibers activated

227.5 210.0 202.5

440.0 425.0 392.5

477.5 452.5 432.5

672.5 657.5 607.5

Table 5. The natural frequencies (Hz) measured with random excitation for pre-tension of 31.14 N/fiber Mode

1

2

3

4

Unactivated Half fibers activated All fibers activated

257.5 275.0 292.5

455.0 452.5 460.0

535.0 590.0 635.0

717.5 737.5 757.5

101

NITINOL-reinforced plates: Part Ill. Dynamic characteristics 1.2 1 i (a) [NIT[AL TENSION: 0.0 N/FIBER [ ALL FIBERS UNACTIVATED

1. 0

I i

-~ 0.8 :~

•

rrl

j

HALF FIBERS ACTIVATED FIBERS ACTIVATED

, ALL

M

;,

i © z

i ~

0.2

~,1

i~,l,

o.o

'

'i "

[//

,'I

',"i

_

0

200

400 600 FREQUENCY (Hz)

800

1000

1.2 ] (b)n~mALTENSION:31.14N/FIBER 1. 0 i

ALL FIBERS UNACT1VATED HALF FIBERS ACTIVATED ALL FIBERS ACTIVATED

,~ 0 . 8

:i I !i

ii

ii I~ 0.6

I

,-d

< 0.4 ~

ili(lii

,:,1

© Z 0.2 0.0 -

0

200

400 600 FREQUENCY (Hz)

800

1000

Fig. 5. Frequency response of an impulsively excited N1TINOL-reinforced plate for pre-tension of (a) 0.0 and (b) 31.14 N/fiber.

Tables 6 and 7 list the effect o f the activation strategy on the amplitude of vibration when the pre-tension is set at 0 and 31.14 N/fiber, respectively. Table 6 indicates that the vibration amplitude decreases for the first four modes of vibration by 12.90070, -4.69°70, 43.76070 and 21.46070 when half the fibers are activated and by 30.02070, 13.93 070, 50.47070 and 18.70070 when all fibers are activated as compared to the case o f a plate with unactivated fibers. When the pre-tension is set at 31.14 N/fiber the amplitude attenuation of first four modes o f vibration is found to be 48.7207o, 30.82°70, 49.89070 and 23.2507o for half

Table 6. Normalized amplitude of vibration measured during random excitation test with pre-tension of 0.0 N/fiber Mode

1

2

3

4

Unactivated Half fibers activated All fibers activated

1.0000 0.8710 0.6998

0.2512 0.2630 0.2162

0.6095 0.3428 0.3019

0.1905 0.1496 0.1549

Table 7. Normalized amplitude of vibration measured during random excitation test with pre-tension of 31.14 N/fiber Mode

1

2

3

4

Unactivated Half fibers activated All fibers activated

1.0000 0.5128 0.2785

0.6918 0.4786 0.3056

0.6607 0.3311 0.1951

0.4120 0.3162 0.1289

102

J. Ro and A. Baz

Mode

Unactivated

Half activated

Fully activated

Fig. 6. Mode shapes of the NITINOL-reinforced plate which were measured by the STAR MODAL package.

activated fibers, and 72.15%, 55.83%, 70.47% and 68.71°70 for all activated fibers. It is evident, therefore, that the activation of the NITINOL fibers is effective in attenuating the plate vibration particularly when the proper pre-tension is used.

3.4. Comparison between theoretical predictions and experimental results The thermal and mechanical finite element models are exercised to predict the thermal and dynamic characteristics of the NITINOL-reinforced plate. These methods utilize the temperature dependence of the modulus of elasticity of the NITINOL-reinforced plates as an input shown in Fig. 8. Such characteristics are obtained experimentally using the Dynamic Mechanical Thermal Analyzer (DMTA) of Polymer Lab. Inc. The temperature-recovery force characteristics shown in Fig. 9, are also used as an input to the mechanical finite element model. Figure 10 shows a comparison between the theoretical and experimental mode shapes when the N I T I N O L fibers are pre-tensioned to 31.14 N/fiber and are unactivated.

N I T I N O L - r e i n f o r c e d plates: P a r t III. D y n a m i c characteristics 1.0 .

.

I

.

.

.

.

.

.

] ]

u~

.

.

.

.

..... ~

.

.

HALF FIBERS ACTIVATED ALL FIBERS ACTIVATED

06 t

© Z

.

(a) INITIAL TENSION: 0.0 N/FIBER -- ALL FIBERS UNACTIVATED

0.8 I i

<

.

103

,i

Oo. I ~

li,'

•

0.0 0

--~800

400 600 F R E Q U E N C Y (Hz)

200

1000

1,0 T . . . . . . . . . . . . . . . . . . . (b) INITIAL TENSION: 31.14 N/FIBER ALL FIBERS UNACTIVATED 0.8 i ..... HALF FIBERS ACTIVATED _ _ ALL FIBERS ACTIVATED

<

06

i

t

ti

J '"

0.4

'

[i

'l ?,

i',

Z

i

i

i'

0.2

0.0 . . . . . . 0

i"

~!1

,,'

I ~ ,", i

~ . . . . r " " ' : • 200 400 600 800 F R E Q U E N C Y (Hz)

~ '

I

i 1000

Fig. 7. F r e q u e n c y r e s p o n s e o f a r a n d o m l y excited N I T I N O L - r e i n f o r c e d p l a t e for pre-tension o f (a) 0.0, a n d (b) 31.14 N / f i b e r .

DRIVING FREQUENCY 1 Hz ~"

4

r~

3 © 2

©

' 0 . 0

I .

.

.

. 30

.

.

. . 60

.

.

TEMPERATURE

. 90

.

.

. 120

i 150

( °C )

Fig. 8. E f f e c t o f o p e r a t i n g t e m p e r a t u r e o n m o d u l u s o f elasticity o f the N I T I N O L - r e i n f o r c e d plate.

J. Ro and A. Baz

104 150

COOL N I~

INITIAL TENSION: 31.14 N 120

z

90

~0

60

30

0

210

40 60 80 TEMPERATURE ( °C )

100

120

Fig. 9. The temperature-recovery force characteristics of the NITINOL fiber.

Mode

Ca)

(b)

Fig. 10. Comparison of the mode shapes between (a) theoretical prediction and (b) experimental measurement.

NITINOL-reinforced plates: Part llI. Dynamic characteristics

105

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Figure l l(a) and (b) shows comparison between the theoretical and experimental natural frequencies of the plate for different initial fiber tensions and different activation strategies. Close agreement is evident between the theoretical predictions and experimental results 5. SUMMARY

The dynamic characteristics of NITINOL-reinforced plates as influenced by different activation strategies and levels of pre-tension of the NITINOL fibers have been presented. The fundamental issues governing the performance of this class of smart composites have been introduced and modeled using finite element method. The accuracy of the developed finite element model has been validated experimentally and against results which are available in the literature. The effectiveness of the NITINOL reinforcement in attenuating structural vibration of composite plates has also been clearly demonstrated. The developed theoretical and experimental techniques present invaluable tools for designing and predicting the performance of NITINOL-reinforced composites that have continuously tunable structural characteristics. Acknowledgements--This work is funded by a grant from Army Research Office (grant number DAAL03-89-G0084). Special thanks are due to Dr Gary Anderson, the technical monitor and Chief of the Structures and Dynamics Branch of ARO, for his invaluable and continuous technical inputs.

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