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Experimental investigation of the dynamic characteristics of carbon fiber epoxy composite thin beams
Composite Structures 33 (1995) 77-86 0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263~8223/95/$9.50 0263.8223(95)00108-5
Experimental investigation of the dynamic characteristics of carbon fiber epoxy composite thin beams Kwang Seop Jeong, Dai Gil Lee & Yoon Keun Kwak Department of Precision Engineering and Mechatronics, Korea Advanced Institute of Science and Technology Kusung-dong, Yusung-gu, Taejon-shi, Korea 305701
The dynamic characteristics of high strength symmetrically laminated carbon fiber epoxy composite thin beams were experimentally investigated in a vacuum chamber equipped with a fiber optic vibrometer and the electromagnetic hammer. The measured dynamic characteristics were compared to those calculated by the macromechanical theory based on the anisotropic plate theory and the Bernoulli-Euler beam theory. IFrom the comparison, it was found that the macromechanical theory could accurately predict the dynamic characteristics of the carbon fiber epoxy composite thin beams when the undirectional properties of the composite material were known.
fiber, matrix, fiber volume fraction, stacking sequence of the reinforced fiber and the interaction between the fiber and matrix,6’7 the dynamic properties of composite materials should be investigated, taking into account these parameters. Several investigators8,9 used the micromechanical analysis based on the rule of mixture to predict the dynamic properties of the composite materials. However, Ni and Adam?’ revealed that the rule of mixture did not predict accurately the major Poisson’s ratio and the directional specific damping capacities of the unidirectional laminated composite material although it predicted well the elastic moduli of the unidirectional laminated composite material. They found that the error increased as the fiber volume fraction increased beyond 60%. However, it has been found that some analytic solutions1’,‘2 based on the macromechanical model predicted well the dynamic properties of the transversely isotropic composite materials. Since the macromechanical analysis cannot be performed without the dynamic properties of the unidirectional composite material, the accu-
INTRODUCTION Carbon fiber epoxy composite materials have been used in advanced structures such as spacecraft, aircraft, automobile transmission shafts and robot structures because of their high specific stiffness and high specific strength.rT2 Since it has also higher material damping than conventional metals, the structure composed of the composite material not only absorbs the noise and vibration of the structure but also improves the settling time and the fatigue life.3 In order to utilize these beneficial properties, many efforts4,5 were attempted to substitute the carbon fiber epoxy composite material for conventional steel or aluminum in advanced structures. As carbon fiber epoxy composite material is being widely employed in advanced structures, its properties, especially dynamic properties, are often specified in the design stage of composite structures. Since the dynamic: properties, such as the natural frequency and damping of fiber reinforced composite materials are dependent on the mechanical properties of the reinforcing 77
78
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
rate measurement of the mechanical properties of the unidirectional composite is indispensable for the macromechanical analysis. Therefore, in this work, an improved test method was devised to obtain reliable experimental data for the macromechanical analysis. In order to obtain accurate dynamic properties of the specimen, the specimen must be excited in a vacuum without any constrained force around the specimen and the accurate non-contact sensors must be used to avoid underestimating the natural frequency due to the sensor mass and overestimating the material damping due to the friction between the sensor and the specimen. Although there have been several experimental investigations on the measurement of the dynamic properties of composite materials, 13,I4 studies that employed both non-contact sensors and a vacuum environment are rare. Since the recently developed fiber optic laser vibrometer system has higher accura@ than the conventional accelerometer, in this work it was employed to measure the dynamic properties of the composite material without inducing any constraint force in the specimen. The measurement was performed in the 10B3 vacuum chamber to eliminate air damping. The high strength carbon fiber epoxy composite material was cured by the autoclave vacuum bag degassing process. The fiber volume fraction of the specimen was measured by burning out the resin of the composite material. The dynamic properties of the specimen were measured with respect to the aspect ratio of the specimen and the stacking sequence. After measuring the dynamic properties, a macromechanical analysis was performed to compare the experimental and calculated results.
THEORETICAL BACKGROUND Beam theory There are two major beam theories, i.e., the Bernoulli-Euler and the Timoshenko beam theories. The latter includes both the shear deformation effect and rotary inertia in the analyis of lateral vibration. Since the composite material usually has a large ratio of longitudinal modulus to the shear modulus (E1/Gr2), the Timoshenko beam theory is more suitable for analysis, although it is more complicated for
composite materials. Dudek16, l7 compared the two moduli calculated by the Bernoulli-Euler and the Timoshenko beam theories and revealed that both beam theories gave almost the same results for composite materials with E1/G12=25 when the ratio of length to height (L/H) was larger than 100. Therefore, in this work, specimens whose L/H was larger than 100 were tested to compare the test results to those calculated by the simple Bernoulli-Euler beam theory. When the densities of the fiber and resin of the composite material are known, the fiber volume fraction can be determined by measuring the weight and volume of the composite material. When they are not known, the fiber volume fraction must be determined by removing the resin from the composite material. In this work, the fiber volume fraction was determined by both methods to estimate the void fraction of the specimens. If pf and pm represent the densities of the fiber and the matrix, respectively, the fiber volume fraction uf and the density of the composite material pC are expressed by the following two equations when there is no void. Wf JPf
“f=wflpf+
(1 -wf)/pm
(1)
1 PC= wflpf+(l-wf)Ipm
(2)
where wf is the fiber weight fraction of the composite material. The void volume fraction of the composite u, is calculated by comparing the theoretical density to the actual density of the composite material as follows. u = PC-Pa 0 PC
(3) \ ,
where pc is the theoretical density of the composite without void and pa is the actual density of the composite with void, which is experimentally determined using the following equation:
where pw is the density of the distilled water, W, the weight of the specimen in the air, and W,
Dynamic characteristics of composite beams
the weight water.
of the
specimen
in the
distilled
Effective flexural modulus Figure 1 shows both the laminate coordinate axis (1, 2, 3) and the ply coordinate axis (x,y,z) with the variables for the symmetrically laminated composite bealm, in which Ok represents the ply angle of the k-th ply, hk and hk_-l the upper and lower distances from the mid-plane, respectively and tk( =hk - hk_ 1) the thickness of the k-th ply. The effective flexural modulus of the composite laminate is expressed by the anisotropic plate theory” as follows.
79
where AU is the dissipative energy per stress cycle, U the maximum strain energy during a stress cycle, q the loss factor, 5 the damping ratio, 6 the logarithmic decrement, Q the quality factor, and # the phase angle between the sinusoidal strain vector and the sinusoidal stress vector in the linear system. The strain energy in the k-th ply of the composite material can be divided into three parts such as the longitudinal, transverse and longitudinal-transverse shear (or simply shear), that originate from the strain energies corresponding to a~& ~$8: and r&&,, respectively.6Y’2 The three components of the strain energy dissipation AU,“, AU,k and AU& are expressed as follows: L
AU,“=
(5)
hk
J[ J 0
where dII is the component of the flexnral compliance of the laminate beam.
1
7
AU;=
dz dL
$yt#w
dz dL
Specific damping capacities Many researchers have used different terms to represent the material damping. Among them, the specific damping capacity (S.D.C.) $ and other terms are related as follows.” -27~~=~5=:26=271Q-~=27r
Fig. 1.
Coordinates
tan $I (6)
1 1
~~CJ~C;W
hk-,
(7)
where t,GLand tiT represent the longitudinal and transverse specific damping capacities, respectively, in the flexural vibration mode, $LT the longitudinal-transverse shear specific damping
for the symmetric
laminated
composite
beams.
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
80
capacity in the torsional vibration mode, L the length of beam and w the width of beam. If the composite beam is composed of N plies, the components of the strain energy dissipation are expressed as follows: N/2
AU,=2
+(2&&6)){(2~k~kb
c AU,” k=l
-(m~-n~)dl6)(h~-h~-l)}]
N/2
AU,=2
(8)
1 AU; k=l N/2
AU,=2
c
-2mknkdl2
AU&.
(mX.‘cos
ok,
nk=sin &)
where Qii and dij represent the elastic moduli and flexural compliance of the laminated composite beam, respectively.
k=l
Then, the total strain energy dissipation the beam is expressed as follows: AU=AU,i-AU,+AU,. The total strain energy U of the laminated posite beam is expressed as follows.
+c&,E&) dz
1
dL
AU of (9) com-
EXPERIMENTAL INVESTIGATION A high strength carbon fiber epoxy composite prepreg with O-15 mm thickness was used to manufacture the specimens.20 The autoclave vacuum bag degassing process was used to cure the prepreg plies under the cure cycle as shown in Fig. 2. The cured composite plates were cut to the required dimensions using a CNC horizontal milling machine with the diamond wheel. Pure resin specimens were also manufactured to measure the resin density. Before curing the pure resin specimen, voids in the resin were eliminated by placing them in the vacuum chamber. The resin in the compoiste specimens were burned out in an electric furnace for 200 min at 500°C. The electronic balance with 0.1 mg accuracy2’ was used to measure the weights of the original specimens and the remaining fibers. The weight of the specimen was measured both
(10)
Then, the specific damping capacity rj of the laminated composite beam is expressed as follows: (11) where
140
Temperature
120 G 0 100 Et a 2 80 $ E 2
Pressure
0.8
Curve
Curve
I I I
60 40
+d(Q:241 -mknk(Q:dfll
x
{(ddll
20
+Qk22dl2+Q&&.s) +Q&&~+Q%M))
+m~dl2-m,nkd,,)(h~-h~_l)}]
30
(12b)
60
90
120
150
180
Time (min)
Fig. 2.
Cure cycles of composite
beam specimen.
81
Dynamic characteristics of composite beams
in the air and in the 4°C distilled water. Then, the densities of the specimens were calculated by eqn (4). The results of the experiment were summarized in Table 1. The longitudinal modulus EL, transverse modulus ET and major Poisson’s ratio uLT of the unidirectional carbon fiber epoxy composite specimens were experimentally determined by the testing method of ASTM D3039-76 and the longitudinal-transverse shear (or simply shear) modulus GLT was determined by the testing method of ASTM D4255-83. Ten specimens were tested for each value and the results were statistically analyzed. The test results are summarized in Table 2 i:n which cLu, cTU and zLTU represent the ultimate strengths in the longitudinal, transverse, and longitudinal-transverse shear directions, respectively. Impulse-frequency response in the flexural vibration mode The dynamic test was performed in the vacuum chamber with lop3 torr vacuum capacity to remove air damping effect. Since contact type vibration sensors, such as an accelerometer or a strain gage, induce the effects of the added mass of the sensor and also the contact damping between the sensor and the specimen, in this work a noncontact fiber optic laser vibrometer system was employed to measure the vibration and damping characteristics. The test specimen was suspended by two thin light Table 1. Fiber volume fraction of the USN 150 composite material Density (lo3 kgh3)
Prepreg Composite Fiber Resin
1.56 1.57 1.75 l-21
Volume fraction (%)
Prepreg Composite
63-61 66.36
Void content
Prepreg Composite
0.08 0.27
(%)
Table 2. Uni-directional
EL Pa) Average Standard deviation Coefficient of variation
131.6 1.74 l-32
strings and an impulse signal given to the specimen by an electromagnetic hammer.14,22 Figure 3 shows the apparatus for the flexural vibration test. The longitudinal damping capacity ll/L and the transverse specific damping capacity $T of the unidirectional composite beam were measured by the impulse-frequency response tests in the free flexural vibration mode. Figure 4 shows the dimensions of the symmetric laminated composite beam for the experimental vibration test. If the specimen is suspended by strings in the vacuum chamber, test results are influenced by the nodal position of the specimen, the suspension length, the string material and the specimen width. Watchman and Tettf23 found that the distance between the suspension position and the nodal point influenced the natural frequency and the damping capacity and higher resonant frequency and damping capacity were obtained for the flexural vibration when the suspension positions were located away from the nodes. Therefore, in this work, the specimen was suspended at the two fundamental nodal points (x/,5=0*224 and O-776) in order to obtain the accurate dynamic properties of the fundamental natural frequency. Also, the material for suspension must be sufficiently flexible compared to the stiffness of the specimen in order to minimize the energy loss due to vibration coupling. In this work, several materials for suspension were tested such as cotton, nylon, silk, and cotton wound polyester threads, and finally the silk thread was selected because it was flexible in the transverse direction and stiff in the longitudinal direction. In order to test the effect of the string length on the dynamic properties of the specimen, several different string lengths were tested. From the test, it was found that the effect of the string length was negligible when the suspension length was larger than 0.5 times of the specimen length. Therefore, in this work, the suspension length was selected as 05 times of the specimen length.
engineering constants of the USN 150 composite material
gLu
(MW
1995 54-9 2.76
ET (GPa) 8.2 O-32 3.94
cTu
WW
60.9 2.72 4.47
GLT Pa) 6-12 O-29 3.61
rLTu
@@a)
74-8 3-14 4-19
“LT
O-28 O-01 0.71
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
82
The specimens with different stacking sequences which range from [&O] to [ +90] with the interval of [ + 51 and the three different lengths (450, 396 and 384 mm) were tested. From the Bernoulli-Euler beam theory, the relationship between the natural frequency and the flexural elastic modulus at the n-th vibration mode of the laminated composite beam is given by the following equation.24
where An and fn are the eigenvalue and the natural frequency of the n-th vibration mode, respectively, L the length, p the density, A the cross section area, I the second moment of inertia, and Ef the effective flexural modulus of the beam, respectively. From eqn (13), the effective flexural modulus, Ef of the laminated composite beam is expressed as follows.
(14)
(13)
Response
,
Signal
Bxcitation Signal
t P/C
(4 Fig. 3.
AT
(a)
Apparatus
for the free flexural vibration
Fig. 4.
Geometry
of the laminated
test (a) front view (b) side view.
beam specimen.
Dynamic characteristics of composite beams
The specific damping capacity $ at the n-th vibration mode with the natural frequency f,, is given as follows.
(15) where Afn is the half-power bandwidth at the nth natural frequency fn. If w and H represent the width and thickness of the beam, the second moment of inertia I is expressed as follows. wH3 I=-=12
AH2 (16)
12
where A(=wH) is the cross-section area of the beam. Substituting eqn (16) into eqn (14), the effective flexural modulus Ef can be rewritten as follows. Ef= 1
48n2L4pf, il:H2
48n2 =-pf;Ly It,”
Torsional vibration test The torsional pendulum method (ASTM D4065-$2) was used to measure the longitudinal transverse shear damping capacity tiLT. To clamp the specimen, the duralumin plate taps16 whose lengths and thicknesses were 40 mm and 1.5 mm, respectively, were adhesively bonded with the epoxy adhesive (IPCO 9923).26 Figure 5 shows the apparatus for the torsional vibration test in this work. The noncircular shaft usually warps when it deforms under torsion.27 However, the warping effect on the deformation of the shaft with simple closed cross section is small when the ratio of cross sectional dimension to length of the shaft is small. Gere27 found that the warping effect on the natural frequency of the beam could be neglected even for the thin walled open cross section if the following condition was satisfied. (19)
(17)
where c(=LIH. The effective flexural modulus of the laminated composite bea.m is not dependent on the beam width but dependent on L and a. Since the effect of beam width may be neglected for measuring resonant frequencies2’ when the ratio of length to width of the beam was larger than 10, in this work, the ratios of length to width of the specimen were selected to be larger than 13-9. Then the effective flexural modulus was determined by measuring the fundamental natural frecluency. Since Af=22.324 for the fundamental natural frequency fi of the composite beam wit.h the free-free boundary condition, eqn (17) can be written as follows. 48n2 Ef=- 22.42 pf~L2d:!=0.944pf~L2az.
83
(18)
Since the laser beam from the vibrometer should penetrate into the vacuum chamber through a thick glass, the glass thickness effect on the laser beam sensitivity was tested. However, no effect of the glass thickness that ranges from 2.5 to 5-O mm was detected. Therefore, in this work, 2.5 mm thickness glass was used for the window of the vacuum chamber.
where II is the mode number (n= 1,2,3,. . .), t the typical minimum wall thickness of the cross section, L the length of the shaft, and D the effective diameter of the cross section. If a and b represent the width and thickness of the cross section of the beam, respectively, the effective diameter D is expressed as follows. D=(4ab/n)l”
(20)
The width and thickness of the specimen selected in this work were 25 mm and 1.2 mm (0=6.18 mm), respectively. In this case, the warping effect could be neglected for the calculation of the fundamental natural frequency of the composite beam when the length of the specimen was larger than 319 mm. Therefore, the specimens whose lengths were larger than 348 mm were tested. In order to measure the longitudinal-transverse shear specific damping capacity $LT, the specimen was excited by giving the sinusoidal signal to the magnetic transducer as shown in Fig. 5. The sinusoidal signal was swept until the maximum amplitude of the specimen was obtained. After the resonant frequency was measured by the frequency counter, the power to the magnetic transducer was turned off. As soon as the power was turned off, the decay of the vibration was measured with the laser vib-
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
84
was calculated using the following equation.
rometer system which was interfaced to the personal computer with GPIB-PCIIA card. The logarithmic decrement 6 was calculated as follows.
S=’
i
ln[8,1B,+i]
l/+12=26.
Table 3 shows the measured damping capacities of the unidirectional symmetrically laminated composite beam.
(21)
RESULTS AND DISCUSSION
where 0, and O,.,, represent the angular displacements at the r-th and r+i-th cycles, respectively. The longitudinal-transverse shear specific 12 of the composite beam damping capacity $
Excitation signal
(22)
The engineering constants and specific damping capacities of the unidirectional laminated composite beam that were obtained experimentally
O~-fx-J /
Power
amplifier
(B & K 2706) Laser vibrometer system (Polytec OFV-3000)
Frequency
counter
I
I Response signal
-generator] t,
GPIB-PCIIA (National Instruments) Fig. 5.
Table 3. Uni-directional
: 3
-
P/C
AT
Printer
Apparatus for the torsional vibration test.
specific damping capacities of the USN 150 composite material
L (mm)
H (mm)
LIH
450 396 348
1.2 1.2
375 330 290
*L
(Q/o)
O-78 1.09 1.37
J/T (%)
4-76 6.43 7-49
tiLT
(%)
9.06 6.58 10.56
85
Dynamic characteristics of composite beams
were substituted into the equations of the macromechanical model in order to obtain the effective flexural modulus and specific damping capacity with respect to the stacking angles. The results obtained by the macromechanical model were compared to the experimental results the obtained through impulse-frequency response in the vacuum chamber. Figure 6 shows the experimental results of the fundamental natural frequencies of the symmetric laminated composite beam with respect to the stacking angles in the flexural vibration mode. Figure 7 shows the effective flexural moduli that were experimentally determined and calculated by the macromechanical analysis. From Fig. 7, it was found that the macromechanical analysis predicted the experimentally determined flexural modulus well. Figure 8 shows the specific damping capacities that were calculated and experimentally determined at the fundamental natural fre-
quency of the symmetrically laminated composite beam with respect to the stacking angles in the flexural vibration mode. From the results of Fig. 8, it was found that the calculated results were in good agreement with the experimentally determined. Figure 9 shows the calculated directional specific damping capacities of the composite beam when the aspect ratio L/H was 290. From Fig. 9, it was found that the specific damping capacity of the symmetrically laminated composite beam came largely from the longitudinal specific damping capacity tiX when the stacking angle was around 0”, while, it came largely from the transverse damping capacity I,&,when the stacking angle was around 90”. Figure 10 shows the directional specific damping capacities with respect to the stacking angles, in which the directional specific damping capacities increased as the ratio of length to thickness of the composite beam decreased.
10 9 8 7 6 gel +
4 3
theoretical exoerimental
T
2 1 0
10
20
30
40
Stacking
50
angle
60
70
80
0
90
0
natural Fig. 6. Experimentally obtained fundamental frequencies in flexural mode with respect to the stacking angles.
140
I
17
[eel,,
,~
0
,I,
10
1
20
30
,I
/
40
50
Stacking
Fig. 8.
2 -
Variation
1
,I,
,I
60
angle
0
70
60
90
[+@I,
of II/ with respect angles.
to the
stacking
10
2h
9
m" !
6 7
2 '6
5 4
zE
3
1 4
6 70 56
L
26
2
14
zu
1
0 I-42 0 10
‘8 x
0
20
30
Stacking
Fig. 7.
40
50
angle
60
70
a0
90
[ * 01,
Variation of the effective flexural modulus with respect to the stacking angles.
rz
-I 0
10
20
30
40
Stacking
(Ef)
50
angle
60
70
a0
90
[e 01,
Fig. 9. Variation of the specific damping capacities respect to the stacking angles when LIH=290.
with
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
86 CONCLUSIONS
In this work, the dynamic characteristics of symmetrically laminated high strength carbon fiber epoxy composite thin beams were tested in a vacuum chamber equipped with a fiber optic laser vibrometer system and an electromagnetic hammer. The test results were compared to those calculated using macromechanical theory. From this investigation the following conclusions are made:
6.
7.
8.
9.
10.
(1) The
macromechanical model predicted well the dynamic properties of symmetrically laminated composite thin beams when the ratio of length to thickness of the beam is larger than 290. specific longitudinal-transverse The (2) damping of the torsional vibration mode had the largest value when the stacking angle was [O],, The damping capacity of the composite (3) beam came largely from the longitudinal damping when the stacking angle was around [O],, and came largely from the transverse damping when the stacking angle was around [go],, The maximum specific damping capacity (4) of the composite beam was obtained when the stacking angle was [ +40],S. (5) As the ratio of length to thickness of the composite beam decreased, the specific damping capacity increased.
11.
12.
13.
14.
15. 16.
17.
18.
19.
REFERENCES
20. ;::
1. Tsai, S. W. (ed), Composite Design, 4th ed., Think Composites, Dayton, Paris, and Tokyo, 1988, Section 1. 2. Lee, D. G., Jeong, K. S., Kim, K. S. & Kwak, Y. K., Development of the anthropomorphic robot with carbon fiber epoxy composite materials. Comp. Struct., 25 (1993) 313-24. 3. Saravanos, D. A. & Chamis, C. C., An integrated methodology for optimizing structural composite damping. ASME Mechanics of Plastics and Plastic Composites, AMD 10 (1989) 167-80. 4. Lee, D. G., Kim, K. S. & Kwak, Y. K., Manufacturing of a scara type direct-drive robot with graphite fiber epoxy composite material. Robotica, 9 (1991) 219-29. 5. Oh, H. S., Jeong, K. S. & Lee, D. G., Design and manufacture of the composite flexspline of a har-
23.
24.
25.
manic drive with adhesive joining. Comp. Struct., 28 (1994) 307-14. Adams, R. D. & Bacon, D. G. C., Effect of fiber orientation and laminated geometry on the dynamic properties of CFRP. J. Comp. Mat., 7 (1973) 402-28. Gibson, R. F., Damping characteristics of composite materials and structures. J. Mat. Engng & Performance, 1(1992) 11-20. Hashin, Z. & Rosen, B. W., The elastic moduli of fiber reinforced materials. J. Appl. Mech., 31 (1964) 223-32. Gibson, R. F., Chaturvedi, S. K. & Sun, C. T., Complex moduli of aligned discontinuous fiber reinforced polymer composite. J. Mat. Sci., 17 (1982) 3499-509. Ni. R. G. & Adams, R. D., A rational method for obtaining the dynamic mechanical properties of laminae for predicting the stiffness and damping of laminated plates and beams. Comp., 15 (1984) 193-9. Gibson, R. F., Dynamic mechanical behaviour of fiber-reinforced composites: measurement and analysis. J. Comp. Mat., 10 (1976) 325-38. Ni, R. G. & Adams, R. D., The damping and dynamic moduli of symmetric laminated composite beams theoretical and experimental results. J. Comp. Mat., 18 (1984) 104-21. Gibson, R. F., Yau, A. & Riegner, D. A., An improved forced-vibration technique for measurement of material damping. Experimental Mech., 6 (1982) 10-14. Suarez, S. A. & Gibson, R. G., Improved impulsefrequency response techniques for measurement of dynamic mechanical properties of composite materials. J. Testing & Evaluation, 15 (1987) 114-21. Polytec Optronics, Inc., Waldbronn, Germany. Dudek, T. J., Determination of the complex modulus of viscoelastic two-layer composite beams. J. Comp. Mat., 4 (1970) 74-89. Dudek, T. J., Young’s and shear moduli of unidirectional composites by a resonant beam method. J. Comp. Mat., 4 (1970) 232-41. Tasi, S. W. & Hahn, H. T., Introduction to Composite Materials, Technomic Publishing Co., Inc., Pennsylvania, 1980, Chapter 5. Lazan, B. J., Damping properties of materials and material composites. Appl. Mech. Rev., 15 (1962) 81-8. Sunkyong Fiber Co. Ltd., Seoul, Korea. A & D Co., Ltd., Tokyo, Japan. Suarez, S. A., Optimization of internal damping in fiber reinforced composite materials. PhD dissertation, University of Idaho Graduate School, 1984, Idaho, USA. Watchman, J. R. & Tettf, W. E., Effect of suspension position on apparent values of internal friction determined by Forster’s Method. Rev Scientific Instruments, 29 (1958) 517-20. Thomson, W., Theory of Vibration with Applications, 3rd ed., Prentice-Hall International, Limited, London, 1988, Chapter 8. Wren, G. G. & Kinra, V. K., An experimental technique for determining a measure of structural damage. J. Testing & Evaluation, 16 (1988) 77-85. IPCO Co., Ltd., Seoul, Korea. Gere, J. B., Torsional vibration of beams of thin walled open sections. J. Appl. Mech., 21 (1954) 381-7.
Experimental investigation of the dynamic characteristics of carbon fiber epoxy composite thin beams Kwang Seop Jeong, Dai Gil Lee & Yoon Keun Kwak Department of Precision Engineering and Mechatronics, Korea Advanced Institute of Science and Technology Kusung-dong, Yusung-gu, Taejon-shi, Korea 305701
The dynamic characteristics of high strength symmetrically laminated carbon fiber epoxy composite thin beams were experimentally investigated in a vacuum chamber equipped with a fiber optic vibrometer and the electromagnetic hammer. The measured dynamic characteristics were compared to those calculated by the macromechanical theory based on the anisotropic plate theory and the Bernoulli-Euler beam theory. IFrom the comparison, it was found that the macromechanical theory could accurately predict the dynamic characteristics of the carbon fiber epoxy composite thin beams when the undirectional properties of the composite material were known.
fiber, matrix, fiber volume fraction, stacking sequence of the reinforced fiber and the interaction between the fiber and matrix,6’7 the dynamic properties of composite materials should be investigated, taking into account these parameters. Several investigators8,9 used the micromechanical analysis based on the rule of mixture to predict the dynamic properties of the composite materials. However, Ni and Adam?’ revealed that the rule of mixture did not predict accurately the major Poisson’s ratio and the directional specific damping capacities of the unidirectional laminated composite material although it predicted well the elastic moduli of the unidirectional laminated composite material. They found that the error increased as the fiber volume fraction increased beyond 60%. However, it has been found that some analytic solutions1’,‘2 based on the macromechanical model predicted well the dynamic properties of the transversely isotropic composite materials. Since the macromechanical analysis cannot be performed without the dynamic properties of the unidirectional composite material, the accu-
INTRODUCTION Carbon fiber epoxy composite materials have been used in advanced structures such as spacecraft, aircraft, automobile transmission shafts and robot structures because of their high specific stiffness and high specific strength.rT2 Since it has also higher material damping than conventional metals, the structure composed of the composite material not only absorbs the noise and vibration of the structure but also improves the settling time and the fatigue life.3 In order to utilize these beneficial properties, many efforts4,5 were attempted to substitute the carbon fiber epoxy composite material for conventional steel or aluminum in advanced structures. As carbon fiber epoxy composite material is being widely employed in advanced structures, its properties, especially dynamic properties, are often specified in the design stage of composite structures. Since the dynamic: properties, such as the natural frequency and damping of fiber reinforced composite materials are dependent on the mechanical properties of the reinforcing 77
78
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
rate measurement of the mechanical properties of the unidirectional composite is indispensable for the macromechanical analysis. Therefore, in this work, an improved test method was devised to obtain reliable experimental data for the macromechanical analysis. In order to obtain accurate dynamic properties of the specimen, the specimen must be excited in a vacuum without any constrained force around the specimen and the accurate non-contact sensors must be used to avoid underestimating the natural frequency due to the sensor mass and overestimating the material damping due to the friction between the sensor and the specimen. Although there have been several experimental investigations on the measurement of the dynamic properties of composite materials, 13,I4 studies that employed both non-contact sensors and a vacuum environment are rare. Since the recently developed fiber optic laser vibrometer system has higher accura@ than the conventional accelerometer, in this work it was employed to measure the dynamic properties of the composite material without inducing any constraint force in the specimen. The measurement was performed in the 10B3 vacuum chamber to eliminate air damping. The high strength carbon fiber epoxy composite material was cured by the autoclave vacuum bag degassing process. The fiber volume fraction of the specimen was measured by burning out the resin of the composite material. The dynamic properties of the specimen were measured with respect to the aspect ratio of the specimen and the stacking sequence. After measuring the dynamic properties, a macromechanical analysis was performed to compare the experimental and calculated results.
THEORETICAL BACKGROUND Beam theory There are two major beam theories, i.e., the Bernoulli-Euler and the Timoshenko beam theories. The latter includes both the shear deformation effect and rotary inertia in the analyis of lateral vibration. Since the composite material usually has a large ratio of longitudinal modulus to the shear modulus (E1/Gr2), the Timoshenko beam theory is more suitable for analysis, although it is more complicated for
composite materials. Dudek16, l7 compared the two moduli calculated by the Bernoulli-Euler and the Timoshenko beam theories and revealed that both beam theories gave almost the same results for composite materials with E1/G12=25 when the ratio of length to height (L/H) was larger than 100. Therefore, in this work, specimens whose L/H was larger than 100 were tested to compare the test results to those calculated by the simple Bernoulli-Euler beam theory. When the densities of the fiber and resin of the composite material are known, the fiber volume fraction can be determined by measuring the weight and volume of the composite material. When they are not known, the fiber volume fraction must be determined by removing the resin from the composite material. In this work, the fiber volume fraction was determined by both methods to estimate the void fraction of the specimens. If pf and pm represent the densities of the fiber and the matrix, respectively, the fiber volume fraction uf and the density of the composite material pC are expressed by the following two equations when there is no void. Wf JPf
“f=wflpf+
(1 -wf)/pm
(1)
1 PC= wflpf+(l-wf)Ipm
(2)
where wf is the fiber weight fraction of the composite material. The void volume fraction of the composite u, is calculated by comparing the theoretical density to the actual density of the composite material as follows. u = PC-Pa 0 PC
(3) \ ,
where pc is the theoretical density of the composite without void and pa is the actual density of the composite with void, which is experimentally determined using the following equation:
where pw is the density of the distilled water, W, the weight of the specimen in the air, and W,
Dynamic characteristics of composite beams
the weight water.
of the
specimen
in the
distilled
Effective flexural modulus Figure 1 shows both the laminate coordinate axis (1, 2, 3) and the ply coordinate axis (x,y,z) with the variables for the symmetrically laminated composite bealm, in which Ok represents the ply angle of the k-th ply, hk and hk_-l the upper and lower distances from the mid-plane, respectively and tk( =hk - hk_ 1) the thickness of the k-th ply. The effective flexural modulus of the composite laminate is expressed by the anisotropic plate theory” as follows.
79
where AU is the dissipative energy per stress cycle, U the maximum strain energy during a stress cycle, q the loss factor, 5 the damping ratio, 6 the logarithmic decrement, Q the quality factor, and # the phase angle between the sinusoidal strain vector and the sinusoidal stress vector in the linear system. The strain energy in the k-th ply of the composite material can be divided into three parts such as the longitudinal, transverse and longitudinal-transverse shear (or simply shear), that originate from the strain energies corresponding to a~& ~$8: and r&&,, respectively.6Y’2 The three components of the strain energy dissipation AU,“, AU,k and AU& are expressed as follows: L
AU,“=
(5)
hk
J[ J 0
where dII is the component of the flexnral compliance of the laminate beam.
1
7
AU;=
dz dL
$yt#w
dz dL
Specific damping capacities Many researchers have used different terms to represent the material damping. Among them, the specific damping capacity (S.D.C.) $ and other terms are related as follows.” -27~~=~5=:26=271Q-~=27r
Fig. 1.
Coordinates
tan $I (6)
1 1
~~CJ~C;W
hk-,
(7)
where t,GLand tiT represent the longitudinal and transverse specific damping capacities, respectively, in the flexural vibration mode, $LT the longitudinal-transverse shear specific damping
for the symmetric
laminated
composite
beams.
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
80
capacity in the torsional vibration mode, L the length of beam and w the width of beam. If the composite beam is composed of N plies, the components of the strain energy dissipation are expressed as follows: N/2
AU,=2
+(2&&6)){(2~k~kb
c AU,” k=l
-(m~-n~)dl6)(h~-h~-l)}]
N/2
AU,=2
(8)
1 AU; k=l N/2
AU,=2
c
-2mknkdl2
AU&.
(mX.‘cos
ok,
nk=sin &)
where Qii and dij represent the elastic moduli and flexural compliance of the laminated composite beam, respectively.
k=l
Then, the total strain energy dissipation the beam is expressed as follows: AU=AU,i-AU,+AU,. The total strain energy U of the laminated posite beam is expressed as follows.
+c&,E&) dz
1
dL
AU of (9) com-
EXPERIMENTAL INVESTIGATION A high strength carbon fiber epoxy composite prepreg with O-15 mm thickness was used to manufacture the specimens.20 The autoclave vacuum bag degassing process was used to cure the prepreg plies under the cure cycle as shown in Fig. 2. The cured composite plates were cut to the required dimensions using a CNC horizontal milling machine with the diamond wheel. Pure resin specimens were also manufactured to measure the resin density. Before curing the pure resin specimen, voids in the resin were eliminated by placing them in the vacuum chamber. The resin in the compoiste specimens were burned out in an electric furnace for 200 min at 500°C. The electronic balance with 0.1 mg accuracy2’ was used to measure the weights of the original specimens and the remaining fibers. The weight of the specimen was measured both
(10)
Then, the specific damping capacity rj of the laminated composite beam is expressed as follows: (11) where
140
Temperature
120 G 0 100 Et a 2 80 $ E 2
Pressure
0.8
Curve
Curve
I I I
60 40
+d(Q:241 -mknk(Q:dfll
x
{(ddll
20
+Qk22dl2+Q&&.s) +Q&&~+Q%M))
+m~dl2-m,nkd,,)(h~-h~_l)}]
30
(12b)
60
90
120
150
180
Time (min)
Fig. 2.
Cure cycles of composite
beam specimen.
81
Dynamic characteristics of composite beams
in the air and in the 4°C distilled water. Then, the densities of the specimens were calculated by eqn (4). The results of the experiment were summarized in Table 1. The longitudinal modulus EL, transverse modulus ET and major Poisson’s ratio uLT of the unidirectional carbon fiber epoxy composite specimens were experimentally determined by the testing method of ASTM D3039-76 and the longitudinal-transverse shear (or simply shear) modulus GLT was determined by the testing method of ASTM D4255-83. Ten specimens were tested for each value and the results were statistically analyzed. The test results are summarized in Table 2 i:n which cLu, cTU and zLTU represent the ultimate strengths in the longitudinal, transverse, and longitudinal-transverse shear directions, respectively. Impulse-frequency response in the flexural vibration mode The dynamic test was performed in the vacuum chamber with lop3 torr vacuum capacity to remove air damping effect. Since contact type vibration sensors, such as an accelerometer or a strain gage, induce the effects of the added mass of the sensor and also the contact damping between the sensor and the specimen, in this work a noncontact fiber optic laser vibrometer system was employed to measure the vibration and damping characteristics. The test specimen was suspended by two thin light Table 1. Fiber volume fraction of the USN 150 composite material Density (lo3 kgh3)
Prepreg Composite Fiber Resin
1.56 1.57 1.75 l-21
Volume fraction (%)
Prepreg Composite
63-61 66.36
Void content
Prepreg Composite
0.08 0.27
(%)
Table 2. Uni-directional
EL Pa) Average Standard deviation Coefficient of variation
131.6 1.74 l-32
strings and an impulse signal given to the specimen by an electromagnetic hammer.14,22 Figure 3 shows the apparatus for the flexural vibration test. The longitudinal damping capacity ll/L and the transverse specific damping capacity $T of the unidirectional composite beam were measured by the impulse-frequency response tests in the free flexural vibration mode. Figure 4 shows the dimensions of the symmetric laminated composite beam for the experimental vibration test. If the specimen is suspended by strings in the vacuum chamber, test results are influenced by the nodal position of the specimen, the suspension length, the string material and the specimen width. Watchman and Tettf23 found that the distance between the suspension position and the nodal point influenced the natural frequency and the damping capacity and higher resonant frequency and damping capacity were obtained for the flexural vibration when the suspension positions were located away from the nodes. Therefore, in this work, the specimen was suspended at the two fundamental nodal points (x/,5=0*224 and O-776) in order to obtain the accurate dynamic properties of the fundamental natural frequency. Also, the material for suspension must be sufficiently flexible compared to the stiffness of the specimen in order to minimize the energy loss due to vibration coupling. In this work, several materials for suspension were tested such as cotton, nylon, silk, and cotton wound polyester threads, and finally the silk thread was selected because it was flexible in the transverse direction and stiff in the longitudinal direction. In order to test the effect of the string length on the dynamic properties of the specimen, several different string lengths were tested. From the test, it was found that the effect of the string length was negligible when the suspension length was larger than 0.5 times of the specimen length. Therefore, in this work, the suspension length was selected as 05 times of the specimen length.
engineering constants of the USN 150 composite material
gLu
(MW
1995 54-9 2.76
ET (GPa) 8.2 O-32 3.94
cTu
WW
60.9 2.72 4.47
GLT Pa) 6-12 O-29 3.61
rLTu
@@a)
74-8 3-14 4-19
“LT
O-28 O-01 0.71
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
82
The specimens with different stacking sequences which range from [&O] to [ +90] with the interval of [ + 51 and the three different lengths (450, 396 and 384 mm) were tested. From the Bernoulli-Euler beam theory, the relationship between the natural frequency and the flexural elastic modulus at the n-th vibration mode of the laminated composite beam is given by the following equation.24
where An and fn are the eigenvalue and the natural frequency of the n-th vibration mode, respectively, L the length, p the density, A the cross section area, I the second moment of inertia, and Ef the effective flexural modulus of the beam, respectively. From eqn (13), the effective flexural modulus, Ef of the laminated composite beam is expressed as follows.
(14)
(13)
Response
,
Signal
Bxcitation Signal
t P/C
(4 Fig. 3.
AT
(a)
Apparatus
for the free flexural vibration
Fig. 4.
Geometry
of the laminated
test (a) front view (b) side view.
beam specimen.
Dynamic characteristics of composite beams
The specific damping capacity $ at the n-th vibration mode with the natural frequency f,, is given as follows.
(15) where Afn is the half-power bandwidth at the nth natural frequency fn. If w and H represent the width and thickness of the beam, the second moment of inertia I is expressed as follows. wH3 I=-=12
AH2 (16)
12
where A(=wH) is the cross-section area of the beam. Substituting eqn (16) into eqn (14), the effective flexural modulus Ef can be rewritten as follows. Ef= 1
48n2L4pf, il:H2
48n2 =-pf;Ly It,”
Torsional vibration test The torsional pendulum method (ASTM D4065-$2) was used to measure the longitudinal transverse shear damping capacity tiLT. To clamp the specimen, the duralumin plate taps16 whose lengths and thicknesses were 40 mm and 1.5 mm, respectively, were adhesively bonded with the epoxy adhesive (IPCO 9923).26 Figure 5 shows the apparatus for the torsional vibration test in this work. The noncircular shaft usually warps when it deforms under torsion.27 However, the warping effect on the deformation of the shaft with simple closed cross section is small when the ratio of cross sectional dimension to length of the shaft is small. Gere27 found that the warping effect on the natural frequency of the beam could be neglected even for the thin walled open cross section if the following condition was satisfied. (19)
(17)
where c(=LIH. The effective flexural modulus of the laminated composite bea.m is not dependent on the beam width but dependent on L and a. Since the effect of beam width may be neglected for measuring resonant frequencies2’ when the ratio of length to width of the beam was larger than 10, in this work, the ratios of length to width of the specimen were selected to be larger than 13-9. Then the effective flexural modulus was determined by measuring the fundamental natural frecluency. Since Af=22.324 for the fundamental natural frequency fi of the composite beam wit.h the free-free boundary condition, eqn (17) can be written as follows. 48n2 Ef=- 22.42 pf~L2d:!=0.944pf~L2az.
83
(18)
Since the laser beam from the vibrometer should penetrate into the vacuum chamber through a thick glass, the glass thickness effect on the laser beam sensitivity was tested. However, no effect of the glass thickness that ranges from 2.5 to 5-O mm was detected. Therefore, in this work, 2.5 mm thickness glass was used for the window of the vacuum chamber.
where II is the mode number (n= 1,2,3,. . .), t the typical minimum wall thickness of the cross section, L the length of the shaft, and D the effective diameter of the cross section. If a and b represent the width and thickness of the cross section of the beam, respectively, the effective diameter D is expressed as follows. D=(4ab/n)l”
(20)
The width and thickness of the specimen selected in this work were 25 mm and 1.2 mm (0=6.18 mm), respectively. In this case, the warping effect could be neglected for the calculation of the fundamental natural frequency of the composite beam when the length of the specimen was larger than 319 mm. Therefore, the specimens whose lengths were larger than 348 mm were tested. In order to measure the longitudinal-transverse shear specific damping capacity $LT, the specimen was excited by giving the sinusoidal signal to the magnetic transducer as shown in Fig. 5. The sinusoidal signal was swept until the maximum amplitude of the specimen was obtained. After the resonant frequency was measured by the frequency counter, the power to the magnetic transducer was turned off. As soon as the power was turned off, the decay of the vibration was measured with the laser vib-
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
84
was calculated using the following equation.
rometer system which was interfaced to the personal computer with GPIB-PCIIA card. The logarithmic decrement 6 was calculated as follows.
S=’
i
ln[8,1B,+i]
l/+12=26.
Table 3 shows the measured damping capacities of the unidirectional symmetrically laminated composite beam.
(21)
RESULTS AND DISCUSSION
where 0, and O,.,, represent the angular displacements at the r-th and r+i-th cycles, respectively. The longitudinal-transverse shear specific 12 of the composite beam damping capacity $
Excitation signal
(22)
The engineering constants and specific damping capacities of the unidirectional laminated composite beam that were obtained experimentally
O~-fx-J /
Power
amplifier
(B & K 2706) Laser vibrometer system (Polytec OFV-3000)
Frequency
counter
I
I Response signal
-generator] t,
GPIB-PCIIA (National Instruments) Fig. 5.
Table 3. Uni-directional
: 3
-
P/C
AT
Printer
Apparatus for the torsional vibration test.
specific damping capacities of the USN 150 composite material
L (mm)
H (mm)
LIH
450 396 348
1.2 1.2
375 330 290
*L
(Q/o)
O-78 1.09 1.37
J/T (%)
4-76 6.43 7-49
tiLT
(%)
9.06 6.58 10.56
85
Dynamic characteristics of composite beams
were substituted into the equations of the macromechanical model in order to obtain the effective flexural modulus and specific damping capacity with respect to the stacking angles. The results obtained by the macromechanical model were compared to the experimental results the obtained through impulse-frequency response in the vacuum chamber. Figure 6 shows the experimental results of the fundamental natural frequencies of the symmetric laminated composite beam with respect to the stacking angles in the flexural vibration mode. Figure 7 shows the effective flexural moduli that were experimentally determined and calculated by the macromechanical analysis. From Fig. 7, it was found that the macromechanical analysis predicted the experimentally determined flexural modulus well. Figure 8 shows the specific damping capacities that were calculated and experimentally determined at the fundamental natural fre-
quency of the symmetrically laminated composite beam with respect to the stacking angles in the flexural vibration mode. From the results of Fig. 8, it was found that the calculated results were in good agreement with the experimentally determined. Figure 9 shows the calculated directional specific damping capacities of the composite beam when the aspect ratio L/H was 290. From Fig. 9, it was found that the specific damping capacity of the symmetrically laminated composite beam came largely from the longitudinal specific damping capacity tiX when the stacking angle was around 0”, while, it came largely from the transverse damping capacity I,&,when the stacking angle was around 90”. Figure 10 shows the directional specific damping capacities with respect to the stacking angles, in which the directional specific damping capacities increased as the ratio of length to thickness of the composite beam decreased.
10 9 8 7 6 gel +
4 3
theoretical exoerimental
T
2 1 0
10
20
30
40
Stacking
50
angle
60
70
80
0
90
0
natural Fig. 6. Experimentally obtained fundamental frequencies in flexural mode with respect to the stacking angles.
140
I
17
[eel,,
,~
0
,I,
10
1
20
30
,I
/
40
50
Stacking
Fig. 8.
2 -
Variation
1
,I,
,I
60
angle
0
70
60
90
[+@I,
of II/ with respect angles.
to the
stacking
10
2h
9
m" !
6 7
2 '6
5 4
zE
3
1 4
6 70 56
L
26
2
14
zu
1
0 I-42 0 10
‘8 x
0
20
30
Stacking
Fig. 7.
40
50
angle
60
70
a0
90
[ * 01,
Variation of the effective flexural modulus with respect to the stacking angles.
rz
-I 0
10
20
30
40
Stacking
(Ef)
50
angle
60
70
a0
90
[e 01,
Fig. 9. Variation of the specific damping capacities respect to the stacking angles when LIH=290.
with
Kwang Seop Jeong, Dai Gil Lee, Yoon Keun Kwak
86 CONCLUSIONS
In this work, the dynamic characteristics of symmetrically laminated high strength carbon fiber epoxy composite thin beams were tested in a vacuum chamber equipped with a fiber optic laser vibrometer system and an electromagnetic hammer. The test results were compared to those calculated using macromechanical theory. From this investigation the following conclusions are made:
6.
7.
8.
9.
10.
(1) The
macromechanical model predicted well the dynamic properties of symmetrically laminated composite thin beams when the ratio of length to thickness of the beam is larger than 290. specific longitudinal-transverse The (2) damping of the torsional vibration mode had the largest value when the stacking angle was [O],, The damping capacity of the composite (3) beam came largely from the longitudinal damping when the stacking angle was around [O],, and came largely from the transverse damping when the stacking angle was around [go],, The maximum specific damping capacity (4) of the composite beam was obtained when the stacking angle was [ +40],S. (5) As the ratio of length to thickness of the composite beam decreased, the specific damping capacity increased.
11.
12.
13.
14.
15. 16.
17.
18.
19.
REFERENCES
20. ;::
1. Tsai, S. W. (ed), Composite Design, 4th ed., Think Composites, Dayton, Paris, and Tokyo, 1988, Section 1. 2. Lee, D. G., Jeong, K. S., Kim, K. S. & Kwak, Y. K., Development of the anthropomorphic robot with carbon fiber epoxy composite materials. Comp. Struct., 25 (1993) 313-24. 3. Saravanos, D. A. & Chamis, C. C., An integrated methodology for optimizing structural composite damping. ASME Mechanics of Plastics and Plastic Composites, AMD 10 (1989) 167-80. 4. Lee, D. G., Kim, K. S. & Kwak, Y. K., Manufacturing of a scara type direct-drive robot with graphite fiber epoxy composite material. Robotica, 9 (1991) 219-29. 5. Oh, H. S., Jeong, K. S. & Lee, D. G., Design and manufacture of the composite flexspline of a har-
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25.
manic drive with adhesive joining. Comp. Struct., 28 (1994) 307-14. Adams, R. D. & Bacon, D. G. C., Effect of fiber orientation and laminated geometry on the dynamic properties of CFRP. J. Comp. Mat., 7 (1973) 402-28. Gibson, R. F., Damping characteristics of composite materials and structures. J. Mat. Engng & Performance, 1(1992) 11-20. Hashin, Z. & Rosen, B. W., The elastic moduli of fiber reinforced materials. J. Appl. Mech., 31 (1964) 223-32. Gibson, R. F., Chaturvedi, S. K. & Sun, C. T., Complex moduli of aligned discontinuous fiber reinforced polymer composite. J. Mat. Sci., 17 (1982) 3499-509. Ni. R. G. & Adams, R. D., A rational method for obtaining the dynamic mechanical properties of laminae for predicting the stiffness and damping of laminated plates and beams. Comp., 15 (1984) 193-9. Gibson, R. F., Dynamic mechanical behaviour of fiber-reinforced composites: measurement and analysis. J. Comp. Mat., 10 (1976) 325-38. Ni, R. G. & Adams, R. D., The damping and dynamic moduli of symmetric laminated composite beams theoretical and experimental results. J. Comp. Mat., 18 (1984) 104-21. Gibson, R. F., Yau, A. & Riegner, D. A., An improved forced-vibration technique for measurement of material damping. Experimental Mech., 6 (1982) 10-14. Suarez, S. A. & Gibson, R. G., Improved impulsefrequency response techniques for measurement of dynamic mechanical properties of composite materials. J. Testing & Evaluation, 15 (1987) 114-21. Polytec Optronics, Inc., Waldbronn, Germany. Dudek, T. J., Determination of the complex modulus of viscoelastic two-layer composite beams. J. Comp. Mat., 4 (1970) 74-89. Dudek, T. J., Young’s and shear moduli of unidirectional composites by a resonant beam method. J. Comp. Mat., 4 (1970) 232-41. Tasi, S. W. & Hahn, H. T., Introduction to Composite Materials, Technomic Publishing Co., Inc., Pennsylvania, 1980, Chapter 5. Lazan, B. J., Damping properties of materials and material composites. Appl. Mech. Rev., 15 (1962) 81-8. Sunkyong Fiber Co. Ltd., Seoul, Korea. A & D Co., Ltd., Tokyo, Japan. Suarez, S. A., Optimization of internal damping in fiber reinforced composite materials. PhD dissertation, University of Idaho Graduate School, 1984, Idaho, USA. Watchman, J. R. & Tettf, W. E., Effect of suspension position on apparent values of internal friction determined by Forster’s Method. Rev Scientific Instruments, 29 (1958) 517-20. Thomson, W., Theory of Vibration with Applications, 3rd ed., Prentice-Hall International, Limited, London, 1988, Chapter 8. Wren, G. G. & Kinra, V. K., An experimental technique for determining a measure of structural damage. J. Testing & Evaluation, 16 (1988) 77-85. IPCO Co., Ltd., Seoul, Korea. Gere, J. B., Torsional vibration of beams of thin walled open sections. J. Appl. Mech., 21 (1954) 381-7.