The effect of square cut-outs on the natural frequencies and mode shapes of GRP cross-ply laminates

Composites Science and Technology 47 (1993) 91-101

THE EFFECT OF S Q U A R E CUT-OUTS ON THE N A T U R A L FREQUENCIES A N D M O D E SHAPES OF GRP CROSS-PLY LAMINATES S. T. Jenq, G. C. Hwang & S. M. Yang Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan 70101, Taiwan (Received 6 December 1990; revised version received 14 April 1992; accepted 10 June 1992) investigations related to the current study are summarized in the following. Monahan 1 showed the effects of square-cuts on the mode shapes and natural frequencies of a clamped isotropic rectangular plate numerically and experimentally. He applied a finite element program to predict natural frequencies and mode shapes of the flat plates and then verified the results by performing holographic analysis. Rajamani and Prabhakaran 2 investigated the effects of square cut-cuts on the natural frequencies of simply-supported flat symmetric laminated composite plates analytically. In this method, it is assumed that the effect of cut-cuts is equivalent to an external loading on the plate. For larger cut-outs, the authors showed that there is a tendency for the higher modes to interchange or switch for all modulus ratios by varying the fiber orientation except for unidirectional graphite/epoxy with a fiber orientation of 30°. Results are given for square, simply-supported composite plates with centrally located square cut-outs for different modulus ratios. Rajamani and Prabhakaran 3 also reported the forced and free vibration response of clampedclamped anisotropic plates with cut-outs formulated in Ref. 2. The size, shape, and location of the cut-out was expressed as a displacement dependent external loading. Results were given for square, clampedclamped plates with centrally located square cut-outs for different modulus ratios. Good agreement was obtained when results for isotropic plates with cut-outs were compared with available theoretical and experimental results. Walley4 showed the effects of cut-out size on the mode shapes and natural frequencies of clampedclamped quasi-isotropic curved glass fiber reinforced plastic (GRP) panels by using a STAGSC-15 program and holographic analysis. He examined three different sizes of interior cut-outs (2 in x 2 in, 2 in x 4 in, and 4in x 4in) and the tendency for mode shapes to switch for large cut-outs. Cyr 6 continued WaUey's work and reported the

Abstract In this work, effects of square cut-outs on the natural frequencies and mode shapes of cross-ply laminates made of S-glass~epoxy, unidirectional pre-preg tape (Fiberite, Hy-E 9134B) are reported. The first three natural frequencies and mode shapes of cantilever plates (with or without cut-out defects) are investigated both experimentally and numerically. Holographic interferometry and piezo-electric sensing techniques are applied in the tests to record vibration mode shapes and natural frequencies, respectively. A finite element code based on the shear deformation theory of plates is developed to analyze the problem numerically. The test findings and calculated results for the first three natural frequencies agree well. The computed mode shapes and laser holograms resemble each other closely for all cases studied. Effects of the size of cut-outs, the location of defects, and the number of layers removed on the vibration frequencies are reported.

Keywords: vibration, cross-ply laminates, cantilever plates, cut-outs, holography, piezo-electric sensor, GRP INTRODUCTION Composite materials have many advantages as aerospace structural materials which require a combination of high strength and stiffness with low weight. It is becoming increasingly important to understand the static and dynamic characteristics as composite materials become more widely used. As we know that aerospace structures need to be cut in order to satisfy design requirements in many cases, such as for window openings, holes for joints etc., it would be useful to designers if the effects of cut-outs on the dynamic behavior of composite structures could be investigated in detail. Several previously reported

Composites Science and Technology 0266-3538/93/$06.00 © 1993 Elsevier Science Publishers Ltd. 91

S. T. Jenq, G. C. Hwang, S. M. Yang

92

effects of cut-out orientation on the natural frequencies and mode shapes of a curved quasi-isotropic GRP panel. It was found that the 0° cut-out orientation had a significant effect on the panel stiffness while the other-cut-out orientations did not adversely affect the stiffness. In addition, it was also reported that if a large number of elements in the finite element mesh are oriented at an angle other than 0° or 90° , then the computer model was artificially stiffened. These studies have been performed to examine the effects of cut-outs on the natural frequencies and mode shapes of composite and metallic structures. However, most of this work is concentrated on through-thickness cut-outs in composite plates or shells. A limited number of publications has been found on the effect of internal (not a throughthickness) cut-outs on the vibration characteristics of laminated cantilever plates. Therefore, the dynamic behavior of cross-ply GRP cantilever plates containing through-thickness cut-outs or internal (not through-thickness) cut-outs located at various locations in the specimens is studied here. In addition, the size effect of the square cut-outs is also studied and compared with Walley's results. 4 A series of tests have been conducted to examine the vibration mode shapes and natural frequencies by means of holographic interferometry7-9 and piezo-electric sensing techniques, l° Moreover, a finite element code based on the shear deformation theory of plates lj is also developed to analyze the current problem numerically and to check against the test findings. EXPERIMENTAL

SETUP

In all tests, the specimens are made of S-glass/epoxy, unidirectional pre-preg tape (Fiberite, Hy-E 9134B). Table 1 shows the mechanical properties of the pre-preg tape used here. Hot pressing and vacuum bagging have been used to make test samples. The curing cycle consisted of the following steps: (1) uniformly increasing the temperature to 180°C in one hour, (2) maintaining this temperature for two hours, and then (3) air cooling the panel to room temperature. During this process the panels were subject to a constant pressure of 300-350kg/cm 2. Dimensions of the rectangular specimens were 6cm × 17 cm × 1.5 mm and the stacking sequence was

Table 1. Mechanical properties of the unidirectional (0") Sglass/epoxy laminate

E~ E: Gl2 Gi3 vl: p

55.5 GPa 25.9 GPa 7-7 GPa 7.7 GPa 0-26 1 881 kg/m 3

described above, with 7 or 11 layers removed symmetrically with respect to the mid-plane of the plates, were also used in the tests. These specimens were used to study the dynamic response of a cantilever composite plate containing the prescribed cut-outs. Figure 1 shows the geometry and boundary conditions of a test specimen without cut-outs. Ten specimens have been studied in this work. A schematic drawing of a test specimen (run nos 2-10) containing defects is shown in Fig. 2. Figure 3 shows a schematic drawing of samples for additional numerical simulations (run nos 11-16). The first three natural frequencies and mode shapes of the cantilever composite plate (with or without cut-outs) were investigated experimentally. Figure 4 shows the test setup for measuring vibration frequencies with a piezo-electric transducer and related instruments. A function generator and power amplifier (AM501, Tektronic) were used to generate a sinusoidal input signal in order to excite a shaker system. The structural vibration response was then measured by using the piezo-electric transducer. When resonance occurred, the corresponding fundamental frequencies were then measured. 4'~2 The observed natural frequencies, as shown in Table 2, are the averaged values based on five test results for each mode. The test data were found to be reproducible and the estimated error resulted from measurement is within 1% for those cases studied. Instead of using an accelerometer (weighing 0-65 g, B&K model 4374) available in the laboratory to

[0/90/0/90/0/90/0/90/0/90/0/90/0/90/0]T. The specimen is rigidly clamped at one end. Three cantilever laminated plates containing 3 c m x 3cm square cut-outs (through-thickness cut-outs) located 3.5, 8.5 or 12.5cm away from the free end were studied. In addition, a set of specimens containing 3cm × 3 c m square internal defects (not throughthickness cut-outs), located at the specific positions

Fig. 1. Geometry and boundary conditions of test samples (thickness 1.5 mm).

Effect of square cut-outs on GRP cross-ply laminates

cut-o~

$pc~llen

com~.i speclme cut 7 layers

cut 7 layers

cut 7

layers

,

q.- 3~ cut 11 layers

3

cut 11 layel's

28.

I

hole

through

cut

hole

through

cut

throughhole

cut

Fig. 2. Schematic drawing of test specimens (the area of cut-out is 3 cm x 3 cm).

I throughhole cut

~

~

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cut

2cm

4~ I; ,-4

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Fig. 3. Schematic drawing of samples for additional numerical simulations.

4. Schematic drawing of vibration measurement.

frequency

measure vibration natural frequencies, the authors used a piezo-electric transducer (weight 0.51 g) to measure the vibration response to specimens because the effect of the mass of the sensor on the structural dynamic response was minimized if the piezo-electric transducer was used. In order to ensure the accuracy of the signals measured, the piezo-electric transducer and accelerometer were mounted on a shaker system (B&K model 4809) and the corresponding dynamic signals compared. Figures 5(a) and (b) show the dynamic response of this shaker, excited at a frequency of 40.8 or 224.4 Hz. They show that the piezo-electric sensor can accurately measure the vibrational frequencies when compared with the corresponding response measured from a B&K accelerometer. Moreover, the influence of the mass of this piezo-electric sensor on measuring the vibrational response of the composite laminate is insignificant because the mass of the transducer is much less than that of the specimens (the ratio between the mass of a piezo-electric sensor and that of the specimen (no. 1) is 1.8%). Holographic interferometry techniques were also applied in the experiments to record vibration mode shapes. A 10 mW He-Ne laser was used as the light source, the incident light being divided by a beam splitter (50%) into the reference light and object light. A spatial filter and pin-hole (10/am) were also used to diffuse laser light on the specimen. The light intensity of the object beam and reference beam were tuned to a ratio of 1:4 in order to produce a clear hologram. When the vibration natural frequency was found, the holographic image was also recorded on film (Agfa-Gevaert 10E-75). The exposure time was about 10 s for this type of film. This film was developed in Kodak solution D-19. Figure 6 shows a complete schematic drawing of the test setup for measuring vibration mode shapes and frequencies. Selected time-averaged holographic pictures for current tests are shown in Figs 7 and 8. Fringe patterns shown in these pictures represent the contour lines of equivalent displacement field and the nodal line can also be observed.

S. T. Jenq, G. C. Hwang, S. M. Yang

94

Table 2. The computed and measured values of natural frequency for the first three modes for a GRP cantilever plate with or without cut-out Run no.

Mode 1 (Hz)

Mode 2 (Hz)

FEM result

Test result

FEM result

Test result

FEM result

Test result

1

41 "38

173"38

41"08

3

41"68

4

43"27

5

39"95

6

41-54

7

44.54

8

34.46

9

39.51

10

45.57

185"43 (6"5%)" 184"16 (6"5%) 185"19 (6"5%) 187"79 (6"9%) 175"67 (4.1%) 174.29 (2.3%) 177-78 (1.7%) 166.78 (5.7%) 169.49 (4.9%) 191.57 (9.9%)

259"36

2

11 12 13 14 15 16

40.77

41 "34 (0"1%)" 43'10 (4"7%) 44"80 (6"9%) 44"94 (3"7%) 41" 14 (2.9%) 43.92 (5.4%) 46.04 (3.3%) 33.58 (2.6%) 41.09 (3.8%) 47.35 (3.8%) --

36.64 38-44 28.58 43.10 49.78

------

240"96 (7"6%)" 251"89 (4"6%) 255'59 (4"8%) 242"72 (7%) 243-61 (8.8%) 250-17 (6.6%) 242.13 (6.6%) 238.78 (7-2%) 228.00 (6.2%) 234.74 (4-1%) ---

a Error (%) = I(OJf¢omputed)-- (3)( . . . . . . .

172"28 173"12 174"82 168"42 170-24 174-82 157-26 161.09 172.54 168.29 152.89 166-83 145.14 172-45 176.71 d))/('O( .......

2.0 solid line : occelerorneter 224..5

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O.B

2 0 ~

(?.2

-

-0.4.

-1.Q 0.0

0.14,

/

I

0.B

'

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263"49 267"76 259"81 264"98 266.80 258.23 255.97 242-05 244.47 249.81 236.11 260.22 240-05 253"92 227.04

--

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MODELING

Hz

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1.4

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\~/

/

7.S

Time (E-02 sec)

A dynamic shear deformation theory of plates" based on small strain and linear stress/strain assumptions has been used to solve this problem numerically. The transverse stress component, oz, is neglected in the analysis. Consider a laminate consisting of N thin orthotropic layers, each layer having constant thickness, and the ith layer (i = 1, 2 , . . . , N) being oriented at an arbitrary angle, Ore, with respect to the plate coordinate axes. The xy plane coincides with the

(a) 3 solid line : accelerometer 2-

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da3hed line : piezo transducer ¢0.82 HZ

{functi°n

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-2

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x.

f

/ ~ spatial.~l b. ~...~fflter

He-Ne laser

sec)

(b) Fig. 5. The comparison of vibration response when using a piezo-electric transducer and an accelerometer (B&K model 4374).

GPIB oscil )scope Fig. 6. Schematic drawing of experimental setup.

Effect of square cut-outs on G R P cross-ply laminates Mode 1 computed test result result

Mode 2 computed test result result

95

displacements; and q~x and q~y denote the rotations in the xz and yz planes due to bending. From Hamilton's principle, i.e:

Mode 3 computed test result result

f t, 6L dt = 0

(2)

o

where 6L is the first variation of the Lagrangian, the equations of motion for plates can be derived as:

aNl + aN6= l, ii + i2~ x ax ay aN6 aN2 + =t, iJ + 12q;y ax ay

Fig. 7. Computed vibrational mode shapes and measured holograms for the first three modes of a cantilever composite plate (without cut-out). mid-plane of the plate. On the basis of the assumptions described, the displacement field can be expressed as:

where t is the time and ul, u2, and u3 denote the displacements of a point along x, y, and z directions, respectively; u, v, and w are the associated mid-plane

(3)

OX F---~y - Q ' = I2ii +13~x

aM2

Here 11, 12, and 13 are the normal, couple normal-rotary, and rotary inertia coefficients, and q represents the transverse distributed force. Ni, Qi, and Mi are stress and moment resultants given by the following equations:

(N. M~) = Mode 2 computed test result result

q + I I fi,

(1)

u3(x, y, z, t) = w(x, y, t)

Mode 1 computed test result result

=

aM, aM6

ul(x, y, z, t) = u(x, y, t) + Zd~x(X, y, t) u2(x, y, z, t) = v(x, y, t) + Z~y(X, y, t)

OQ---3+ OQ2 oy ax

Mode 3 computed test result result

I?

(1, z)oi dz

and

(Q~, Q2)

hi2

~

h12

=

(05, o,) dz

(4)

a-hi2

We note that i = 1, 2 . . . . . and the stress component is denoted by oi (i.e. o l = O x , (rE=Oy, 04=aye, Os=Oxz, and o6=oxy). In addition, anisotropic stress]strain relationships of an individual layer m are of the form:

o; (a)

Mode 1 computed test result result

Mode 2 computed test result result

Mode 3 omaputed test result result

=

(5)

where o' and e; are referred to as lamina material coordinates, and c~-~m) are the elastic constants of the mth lamina. Indices i and j range from 1 to 6. After transforming eqn (5) to stresses in the plate coordinates (x, y, z), we may derive the constitutive equation as: 13 equation as:13 o,. = O{.~)p ,~,, ~,

(6)

where Q~."), oi, and ei are the material coefficients, stresses, and strains in the mth lamina with respect to the plate axes. A four-node finite element formulation is applied to solve this cantilever plate and the governing equation for each element can be written in matrix form as:

(b) Fig. 8. T h e c o m p u t e d a n d m e a s u r e d first t h r e e m o d e s h a p e s f o r s p e c i m e n s ( a ) n o . 9 a n d ( h ) n o . 10.

[Mq[~'°l + [ r ° l [ X °1 = [e°]

(7)

where [M ~] and [K ~] are the element mass and

S. T. Jenq, G. C. Hwang, S. M. Yang

96

Table 3. Comparison of non-dimensional frequencies (~°) for a square simply-supported isotropic plate m

1 1 1 2 3 1

n

1 2 3 3 3 5

3D linear elasticity solution

Mindlin's thick plate solution

2x 2 meshh

Classical plate theory

FEM 4 x 4 mesh'

5.780 13-805 25-867 32-491 42.724 57.476

5.767 13.755 25.700 32-230 42.302 56.758

5.920 15.252 28-133 33.057 49.723 74-041

5.973 14-934 29-867 38-829 53.868 77.652

5.774 13.881 26-760 34.022 47.714 68.864

toa2(p/Eh2), a/h = 10, and Poisson's ratio is 0.3. b Reddy.15 c Present work. a ~ =

stiffness matrix, respectively; and [F e] and [X e] are the force and displacement vectors. Note that ['] represents the time derivative of [ ] and [X e] can be written as [u e, v% w e, ~ , q~y] e T. The global governing equation can be obtained after assembly of the equation of motion for each element. ~4 For a vibration system, the corresponding characteristic equation can be expressed as: I[K] - to2[M]l = 0

(8)

where to2 represents the system's eigenvalue, and to is the natural frequency of the plate. The vibration mode shape of a cantilever plate can then be determined after the corresponding eigenvector is found. The cantilever plate boundary conditions for the current problem are represented in Fig. 1. Note that the clamped boundary conditions may not always exactly represent the physical situation used in the tests, and these assumptions may therefore lead to uncertainties in the numerical results compared with the test findings. Previously reported work has been used to check the present computed results in order to ensure

numerical accuracy. A comparison of the vibration frequencies between the authors' results and those of Reddy is for a square, simply-supported isotropic plate is shown in Table 3. This table shows that the difference between the current computed results and 3D linear elasticity solutions is within 4-7% for the first four fundamental frequencies. In addition, Table 4 shows the computed vibration frequencies of a cantilever angle-ply laminated plate reported by Crawley. 12 The difference between the present authors' predictions and Crawley's calculated results is within 3-3% for the first three natural frequencies. This means that the current program is capable of predicting the vibration response closely. Moreover, the convergence test is also performed to ensure the accuracy of the authors' numerical solutions. Table 5 shows the calculated first three fundamental frequencies based on 24 (3 x 8), 408 (12 x 34), and 504 (14 x 36) meshes of a specimen without cut-outs and the corresponding plot of the calculated frequencies versus the number of elements is also shown in Fig. 9. On the basis of these results, the mesh size chosen here is 0.5 cm × 0-5 cm (408 elements in a specimen

Table 4. Comparison of frequencies (Hz) for a cantilever angle-ply laminated plate tz

[0/0/30/-30]s laminate

Present FEM prediction

65.37

65.45 (0.12%)" 138-10 (0-43%) 421.76 (3.2%) 542.66 (3.l%) 620.55 (6%)

137.50 408.30 525.60 588.30

[0/45/-45/90]s laminate 55.58 175.40 345.30 591.80 820.10

Present FEM prediction 55.64 (0-11)" 175.90 (0-28%) 357.14 (3.3%) 608.95 (2-8%) 865.79 (5.3%)

[45/-45/-45/45]s laminate 31.90 191-30 228.20 565.30 708-30

Present FEM prediction 31.82 (0.25%)" 196-91 (2.8%) 228-41 (0-1%) 614.54 (8%) 724.95 (2.3%)

a The value in the bracket represents the error between this analysis and that of Crawley~2.

Effect of square cut-outs on GRP cross-ply laminates

97

Table 5. The calculated first three fundamental frequencies for the cross-ply GRIP cantilever plate without cut-outs (i.e. specimen 1) based on three different types of meshes

Computed mode 1 Computed mode 2 Computed mode 3 frequency frequency frequency (Hz) (Hz) (Hz)

No. of mesh

24 (3 x 8) 408 (12 x 34) 504 (14 x 36) 3O0

41.47 41.38 41.38

269-11 259-36 259-30

Figs 10(a)-(c) is 3 x 3 cm 2. The solid and dashed lines in these figures represent the computed results, while the points shown in the figures represent the test findings. Figures 10(a)-(c) show that the first three

mode 3

250 mode 2

200 ~

166.72 173.38 173.38

150 100 50 • 0

mode 1 •

!

0

400'

l

s

.

,

300"

i

100 200 300 400 500 600 no. of elements

Fig. 9. The computed frequencies versus number of meshes used in the program for a specimen without cut-out (i.e. run no. 1).

12 200'

rrgxte 3

"r • -.. mode 2 . . . . . . . . . . -t. . . . . . e.. . . . . It

4~ 100'

mode I

0T 0

without cut-out) in order to optimize the accuracy of the computed results, computing time, and the available run-time memory in the authors' workstation. RESULTS

AND

DISCUSSION

The first three vibration mode shapes of a cantilever plate without defect (specimen 1) are shown in Fig. 7. Both experimental and numerical results show that the first three modes are governed by the bending and torsional mechanisms. Selected vibration mode shapes obtained from the computed results and hologrr, .s are shown in Figs 8(a) and (b) for specimens 9 and 10. These results show that the mode shapes of specimens resemble each other closely in the current study. A number of specimens made of S-glass/epoxy have been studied here. Table 2 summarizes the natural frequencies of the first three modes obtained from tests and code simulations. This table also reveals that natural frequencies of a cut-out specimen can be either higher or lower than those found in an uncut specimen. It is also found that the natural frequencies of the plate may depend upon the location of cut-out, the amount of mass removal and the stiffness degradation due to cut-out. Figures 10(a), (b), and (c) show the relationship between the number of removed layers and the natural frequencies when the defect is located 4-5, 8.5, or 13.5 cm away from the clamped end, respectively. The area of cut-out for these specimens presented in

m |

|

,

5

10

15

no. of layers removed

(a) 400 mode 3

-- 30O

~

i

mode 2

200

. . . . . . . . .

100

-t

. . . . .

O. . . . . .

•

_-

•

mode 1 _-

oI 0

5

10

15

no. of layers removed

(b) 400 A 300

mode 3

~200

g. 100" mode I i

i

5

10

15

no. of layers removed

(c) Fig. 10. The computed and measured natural frequencies of a composite cantilever plate with or without cut-out, when the 3 cm x 3cm defect is located (a) 4-5, (b) 8-5, or (c) 13.5 cm away from the clamped end.

98

S. T. Jenq, G. C. Hwang, S. M. Yang

vibration frequencies tend to decrease if the number of layers containing cut-out is increased, except that the vibration frequency of mode 1 increases when the defect is located close to the free end. This means that the effect of the degradation of structural stiffness is important compared to the effect of mass removal due to cut-out. However, if the defect is located close to the free end of the cantilever plate, the effect of the stiffness degradation becomes less important than that of the mass removal. Therefore, the mode 1 shown in Fig. 10(c) tends to increase as the removed layers are increased. This behavior can be explained by using a static load applied to the cantilever beam and it is found that the transverse deflection increases (i.e. the plate becomes less stiff) when the defect is moved toward the clamped end. When the higher mode is considered, the effect of rotary inertia, higher order shear corrections, and the variation of the dynamic moduli 12 may also play a role and result in decreasing the vibration frequencies. Moreover, Figs l l ( a ) - ( c ) show the difference of the computed first three natural frequencies between an intact sample and the pre-cut specimen versus the number of layers removed when the 3 c r u x 3 c m cut-out is located at (a) 4.5, (b) 8-5, or (c) 13-5 cm away from the clamped end. Notice that Figs l l ( a ) - ( c ) are plotted from the numerical results presented in Figs 10(a)-(c). A maximum of 16.7% change of the first natural frequency is found in Figs l l ( a ) to (c), when a 3 c m × 3 c m through-thickness cut-out (i.e. 15 layers removed) is located 4.5 cm from the clamped end. Figure l l ( c ) also reveals that the first vibration frequency increases as the removed layers are increased. When there are seven layers removed from the specimen (i.e. run no. 2, 3, or 4), it shows that the difference of the first three natural frequencies between an intact sample and the specimen with cut-out is within 5%. For the second and third modes, the change is less than 10% for the cases shown in Figs l l ( a ) - ( c ) . Figures 12(a), (b), and (c) present the relation between location of the defect and natural frequencies when there is a 7-, 11-, or 15-layer cut-out defect (with an area of 3 cm × 3 cm) located symmetrically with respect to the mid-surface of the plate. The test results are shown as the points in the figures and the solid and dashed lines represent the computed values. It shows that the vibration frequencies of mode 1 and 2 increase as the defect is moved away from the clamped end because the sample becomes stiffer. However, the frequency of the third mode tends to decrease if the defect is moved away from the clamped end. In addition, Figs 13(a)-(c) present the difference of the computed natural frequences between an intact sample and specimen containing the 3 cm x 3 cm cut-out versus the location of the cut-out when 7, 11, or 15 (through-thickness) layers are

20

run#2

run#5

run#8

7

11

15

;~ -10 -20 no. of layers removed

(a) 20" r u n # 3

run#6

run#9

11

15

10

"~" -10"

mode 2 R rr~le 3

-20 ~ 7

no. of layers removed

(b) 20-

run # 4

-10mode2 . II! mode3 -20 7

run # 7

run # 10

" 11

15

no. of layers removed

(c) Fig. 11. The difference (%) of the natural frequencies between an intact sample and pre-cut (with an area of 3 cm x 3 cm) cantilever plate if the cut-out is located (a) 4.5, (b) 8.5, or (c) 13.5cm away from the clamped end. * Difference ( % ) = [(vibration frequency of specimen with cut-out- vibration frequency of intact specimen)/vibration frequency of intact specimen] x 100.

removed. This shows that the difference increases when the removed layers are increased. A plot of the first three fundamental frequencies versus the size of the through-thickness, square cut-out located in the center of the cantilever plate is shown in Fig. 14. The points presented in the figure represent the test finding of run nos 1 and 9. The lines shown in this figure represent the calculated frequencies for specimens containing 2 c m x 2 c m , 3 cm × 3 cm, and 4 cm × 4 cm cut-outs. It is found that the fundamental frequencies decrease as the size of cut-out is increased. This result is similar to that reported by Walley 4 for the clamped-clamped

Effect o f square cut-outs on G R P cross-ply laminates 400 300 200"

2°t

mode 3 ~---1" .... mode 2 t ....

IL. . . . .

o

|

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i

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10

15

20

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1

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4.5

20mode 3 ~e

100

~

0

2 . . . . .

run#5

mn#6

[] model • mode2 [] mocle 3

-10

u

w

w

5

10

15

-20 4.5

20

8.5

(b) 20'

400'

~ "IV - - - |

200'

mode 2

100-

mode 1

0

0

run#8 mn#9 [] model! 10' [] mode2 [] mode 3 !

mode 3 ~- -

13.5

distance from root (cm)

(b)

300:

mn#7

0

-4'

d i s t a n c e f r o m r o o t (cra)

A

13.5 (cm)

10

mode 1

0

mode 2

(a)

400"

O. . . . .

8.5

distance from root

(a)

200

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d i s t a n c e f r o m r o o t (crn)

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o

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u

,

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5

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distance from root (cm)

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13.5

distance from root (cm)

(c)

(c)

Fig. 12. Plots of the natural frequencies versus the location of defect measured from the clamped end, when (a), 7, (b) 11, or (c) 15 layers (through-thickness) are cut.

Fig. 13. Plots of the difference (%) of the natural frequencies between the intact and pre-cut (with an area of 3 cm x 3 cm) cantilever plate versus the location of cut-outs measured from the clamped end, when (a) 7, (b) 11, or (c) 15 layers (through-thickness) are cut. * Difference ( % ) = [(vibration frequency of specimen with c u t - o u t - vibration frequency of intact specimen)/vibration frequency of intact specimen] × 100.

[0°/-45°/45°/90°]s panels (with a 30.48cm (12-in) chord and height, m a d e of g r a p h i t e / e p o x y composite) containing centrally located through-thickness cutouts. Figures 15(a)-(c) present the difference of the calculated vibration frequencies between an intact sample and specimen with through-thickness cut-out (with an area of 2 c m x 2 c m , 3cmx3cm, or 4 cm x 4 cm) versus the location of cut-out m e a s u r e d from the clamped end. Figure 15(c) shows that the change of the first natural frequency is about 30-9% when an intact sample (run no. 1) and a specimen containing a 4 c m x 4 c m through-thickness cut-out located 4.5 cm away f r o m clamped end (run no. 14) are compared. Again, Figs 15(a)-(c) show that the difference increases when the area of cut-out is

increased. For a smaller cut-out, as shown in Fig. 15(a), the difference seems to be limited within 10% for the first three modes. The difference of the c o m p u t e d frequencies between an intact specimen and the pre-cut sample versus the area of through-thickness cut-out is shown in Figs 16(a)-(c). Figure 16(b) shows that the difference is within 15% for the first three modes, when cut-outs are located at the center of the plates. If the cut-out is located close to the clamped end, the m a x i m u m difference of 30-9% is found in Fig. 16(a) for m o d e 1. In addition, a 20.3% difference is also

100

S. T. Jenq, G. C. Hwang, S. M. Yang 300" rr~e 3

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16

area of cut-outs (cm*cm)

Fig. 14. The vibration frequencies versus the area of through-thickness cut-outs located in the center of the plate for run nos 1, 9, 11, and 12. 4O 3o 20

mn#14

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run#9

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(c) Fig. 15. Plots of difference (%) of the natural frequencies between an intact plate and the specimen with throughthickness cut-out versus the location of defect measured from the clamped end, when the area of cut-outs is (a) 2cm×2cm, (b) 3 c m × 3 c m , or (c) 4 c m x 4 c m . * Difference ( % ) = [(vibration frequency of specimen with cut-out- vibration frequency of intact specimen)/vibration frequency of intact specimen] × 100.

i ~

40 30 20 10 0 -10 -20 -30 -40

run # 15

run # 10

run # 16

I

model mode2 m mode3 4

9

16

area of cut-outs (cm*cm)

(c) Fig. 16. Plots of the difference (%) of the natural frequencies between an intact plate and the specimens with through-thickness cut-out versus the area of cut-out (i.e. 2 c m x 2 c m , 3 c m x 3 c m , and 4 c m x 4 c m ) , when the cut-out is located (a) 4.5, (b) 8-5, or (c) 13.5 cm away from the clamped end. * Difference ( % ) = [(vibration frequency of specimen with cut-out- vibration frequency of intact specimen)/vibration frequency of intact specimen] x 100.

observed for the first vibration frequency of specimen run no. 16. The difference of the first mode shown in Fig, 16(a) explains that the vibration frequency continues to decrease as the area of the cut-out is increased. This is due to the fact that the defect is located close to the clamped end and the stiffness degradation seems to be more important than the loss of mass. When the defect is located close to the free end of the cantilever plate, the mass removal becomes more important and affects the first vibration frequency. Figure 16(c) therefore, shows that the first

Effect o f square cut-outs on G R P cross-ply laminates

natural frequency increases as the area of cut-out is increased. The current work extends those results reported by Wally4 and Crawley ~2 to study the effect of various types of cut-outs located at different positions in cross-ply, GRP cantilever plates. The difference between the test findings and the numerical results is within 10% for the cases studied here. In Crawley's work, ~2 the difference between the fundamental frequencies of the calculated and observed results is less than 2% for an aluminum plate (1.06mm (0.0416-in) thickness) without cut-out. He also reported that the difference became larger (less than 15%) when the composite laminates (without cut-out) were studied. Both Crawley's and the authors' own work suggest that the variation of the dynamic moduli appears to affect the natural frequencies because the difference between the test findings and numerical solutions for the samples made of composites (with or without cut-out) is larger than that for a metallic sample. Therefore, a further study to characterize the dynamic properties of composite materials seems to be important. CONCLUSION In this paper, the effect of square cut-out defects on the first three vibration mode shapes and natural frequencies of cross-ply, GRP cantilever plates has been reported from numerical simulations and experimental results. The mode shapes are found to be similar for all cases studied. The difference between the computed natural frequencies and the measured data for all cases studied is within 10%. For a lower vibration mode (such as mode 1 or 2), mass removal and reduction of structural stiffness seem to be the dominant factors to affect structural vibration frequencies. When we study the higher mode (such as mode 3) the effect of variation of dynamic properties of laminate, rotary inertial and transverse shear needs to be considered in addition to the mass removal and the loss of stiffness due to cut-out. From the calculated results for specimens containing 3cm x 3 cm cut-outs, a maximum of 16.7% change of the vibrational frequency between an intact specimen and the sample with cut-out located 4.5 cm away from the clamped end (mode 1 of run no. 8) is found. For the second and third modes, this change is less than I0%, as shown in Figs ll(a)-(c). For samples containing 4cm × 4cm through-thickness cut-outs (located 4.5, 8.5, or 13.5cm away from the clamped end), the maximum difference found in Fig. 15(c) or 16(a) is about 30.9% when the defect is located close to the

101

clamped end (i.e. mode 1 of run no. 14). It is also found that the difference is within 17% for the second and third modes when the area of through-thickness cut-outs ranges from 2 cm x 2 cm to 4 cm x 4 cm.

ACKNOWLEDGMENT The authors would like to thank the National Science Council, R.O.C., for sponsoring this work under Contract No. NSC 82-0401-E006-138.

REFERENCES 1. Monahan, J., Natural frequencies and mode shapes of plates with interior cut-outs. MS thesis, Air Force Institute of Technology, Wright-Patterson AFB, OH, 1970. 2. Rajamani, A. & Prabhakaran, R., Dynamic response of composite plate with cut-outs, Part I: Simply-supported plates. J. Sound & Vib, 54 (1977) 549-65. 3. Rajamani, A. & Prabhakaran, R., Dynamic response of composite plate with cut-outs, Part II: Clampedclamped plates. J. Sound & Vib., 54 (1977) 565-78. 4. Walley, R. A., Natural frequencies and mode shapes of curved rectangular composite panels with interior cut-out. MS thesis, Air Force Institute of Technology, Wright-Patterson AFB, OH, 1985. 5. Almroth, B. O., Brogan, F. A. & Stanley, G. M., Structural analysis of general shells Vol. II, user instructions for STAGSC-1. MSC-D673837, Applied Mechanics Lab., Lockheed Palo Alto Research Lab., Palo Alto, CA, 1981. 6. Cyr, G. J., Effect of cut-out orientations on natural frequencies and mode shapes of curved rectangular composite panels. MS thesis, Air Force Institute of Technology, Wright-Patterson, AFB, OH, 1986. 7. Vest, C. M., Holographic Interferometry. John Wiley, New York, 1979. 8. Listovets V. S., & Ostrovskii, Y. I., Hologram interferometry for vibrational analysis. Soy. Phys. Tech. Phys., 19 (1975) 847-60. 9. Li, Q. B. & Malohn, D. A., Validation of FEM modal analysis using holographic interferometry. Sound & Vib., 22(11) (1988) 24-32. 10. Beckwith, T. G., Buck, N. L. & Marangoni, R. D., Mechanical Measurements. Addison-Wesley, Reading, MA, 1982. 11. Reddy, J. N., Energy and Variational Methods in Applied Mechanics. John Wiley, New York, 1984. 12. Crawley, E. F., The natural modes of graphite/epoxy cantilever plates and shells. J. Composite Materials, 13 (1979) 195-205. 13. Jones, R. M., Mechanics of Composite Materials. Scripta, Washington, DC, 1975. 14. Reddy, J. N., An Introduction to the Finite Element Method. McGraw-Hill, New York, 1984. 15. Reddy, J. N., Free vibration of antisymmetric angle-ply laminated plates, including transverse shear deformation by the finite element method. J. Sound & Vib., 66 (1979) 565-76.