Damping Analysis -Beam With Composite Coat

8/11/2019 Damping Analysis -Beam With Composite Coat

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Composire tructures 32 (1995) 33-380 1995 Elsevier Science Limited

0263-8223(95)00054-2

Printed in Great Britain. All rights reserved0263-8223/95/ 9.50

Vibration and damping analysis of beams withcomposite coats

Ahmed Abd El-Hamid HamadaProduction Engineeri ng M echanical Design Department, Faculty of Engineeri ng, M enouji a Uni versit y, Shebin El-K om, Egypt

Numerical and experimental investigations of the dynamic behavior forcoated laminate composite beams has been presented and discussed in thepresent work. A numerical technique is utilized to compute the eigenparameters of coated laminated composite beams. An attempt to study thevariations in the natural frequencies and damping properties of laminatedcomposite coated beams taking into account different lamina orientation ofcoat and various kinds of core isotropic material (steel and aluminium) isintroduced. The variations of the eigen parameters vs. the code number ofthe coated layer with the use of various types of isotropic material aremeasured by utilizing (the frequency response displayed on) an (FFT)analyzer. The experimental and numerical work is carried out on fourdifferent fiber orientations, aligned longitudinally, transversely, making 45with the fibers direction and randomly oriented. Comparison betweenexperimental and numerical results shows a tight connection between them.

INTRODUCTION

Fiber reinforced composite materials have beenwidely used in applications ranging from lami-nated aircraft wings in the aeronautical industryto the coating on brake pedal assemblies in theautomobile industry. The increasing usage ofcomposite materials as a coating of conven-tional materials in structural applications isattributed to their high strength to densityratios, high resistance to fatigue, low coefficientof friction, and the lotv damping capacities of

the high strength material. On the other handthe use of laminated composite materials as acoating for isotropic materials may well provevaluable for structural purposes such as thereduction of vibrations and noise in sheet metalframes of machines and an increase in the wearresistance at contact surfaces.-5

Coated laminated composites offer the com-bined advantages such that the great stiffnessand strength of usual elastic sheets can be uti-lized in the design to provide the neededstrength and to increase the total internaldamping in the structure.6-8 In the presentwork, a constrained coated isotropic beam with

different lamina orientation of composite coatsof the core with various kinds of isotropic mate-rial has been investigated numerically andexperimentally. The numerical work includesthe computation of the eigenfrequencies ofelastic constrained beams with different codenumbers in terms of the equivalent stiffness andmass parameters. The experimental results infrequency domain with the utilization of FFTanalyzer are listed and represented in variouscurves.

PROBLEM STATEMENT

For a composite coated beam the equivalentstiffness can be recast in the form:

E,I=C 2Ekb Jk ;2 dy, 1)d,2b c

=Tk=O,l.2..( - ),

where E, can be expressed in terms of the on-axis properties of the lamina serial E,l, E22,

33


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34 A. A. El- Hami d H amada

G12 and v12.10

1+- n4

E22(2)

where m = cos 8; n = sin 8.

Similarly the equivalent mass per unit length,can be cast as:

m=C k=C bpuksk k+dy,k 4=2b c

k=O,l,z.,. h(dk+rdk)*(3)

where pk is the density of the k layer, and b isthe width of the beam and mk is the mass perunit length of the Kth layer. Here the distancefrom the neutral axis of the beam to the twosuccessive layers k and k + 1 is given by dk asshown in Fig. 1.

NUMERICAL WORK

With the help of eqns (1) and (3), the naturalfrequencies of the laminated coated beam canbe then expressed in the form:9

i=O,1,2,3... (4)

where Ai is a function of the boundary condi-tions and L is the length of the beam. Theeigen-pairs of the laminated coated beams interms of equivalent stiffness and mass are com-

Cross section at C-C

i

I- b -I

Y

t

I-L

-i

Fig. 1. A laminated composite coated beam.

puted and listed in Tables 3 and 4. Referring tothe mixture role, the mechanical properties ofcomposite unidirectional lamina of continuousfiber glass reinforced plastics are calculated andwith the use of the Pagano-Tsail and Hu1113

models the properties of the random fiber arecalculated:

EC=+ E,+3 ET (5)

Gc= E,+ ET

The mechanical properties of composite unidir-

ectional lamina are calculated using mixturerole, and the numerical results for the glass/polyester composite are listed in Table 1.

EXPERIMENTAL WORK

The test specimen is a laminated coated beamdimensions 300 x 25 mm and thickness 7 mm.The thickness of the coated layer of fiber glassis 1 mm for each side of core beams, Fig. 2. Thefiber volume fraction is 0.54%. Eight specimens

were constructed and manufactured using thehand lay out technique and the raw materialwas obtained from the Arab Company forDeveloped Materials, Maadi, Cairo.

The experimental apparatus is shown in Fig.3. The specimen was mounted in a test rig andexcited by a B&K (8202) impact hammer, witha force transducer type (8200) built into the tipto register the force input.

The excitation signal was fed to the analyzerthrough the charge amplifier (2635) at the endand mid point position of the specimen whereall modes would be excited.

The resulting vibration response was regis-tered by a piezoelectric accelerometer type

Table 1. Elastic moduli of E-glass/polyester unidirec-tional composite lamina with l/f=034

Elastic modulus Resultant usingmixture rule

El1 (GWh2 Wa)

G12 @Pa)v 2

40.00

:::0.25


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Vibr at io n and dampi ng anal ysis of coat ed beams 35

(4374) stuck on the desired measuring point ofthe specimen by using beeswax. The acceler-ometer signals were conditioned in the chargeamplifer and fed to the dual channel signal ana-lyzer (2304). The analyzer, in conjunction with

the fast fourier transform (FFT), gives themathematical connection between time and fre-quency, successively displaying the frequencyresponse spectrum (FRS) and the coherencefunctions, registered in the given frequencyrange as shown in Fig. 4.

MEASUREMENTS OF EIGEN PARAMETERSOF COATED SPECIMENS

To study the effect of fiber lamina orientationof coating and type of core isotropic material(steel and aluminium) on the dynamic behaviorof coated beams, different boundary conditionswere imposed on the specimen under vibra-tional systems mounted on the testing rig. Thespecimen location in the test rig is ensuredusing tightened clamped edges. Eight specimensof steel and aluminium beams are coated withfour different fiber orientations, aligned longitu-dinally, transversely, making 45 with the fiberdirection and randomly oriented. The ampli-

tudes were measured in the normalized form atvarious points. The frequency response spec-trum (FRS) displayed by the analyzer, theaverage damping factor [ is utilized to estimate

Composite coated Core isotropic material

I7mm

f

Fig. 2. Dimensions of coated specimen beam test.

Dot. matrixprinterND-15

Desktop computer

five frequency bands assuming damping linear-ity using half power band width. The dampingfactor < of a particular resonance can be calcu-lated from the width of the resonance peak inthe magnitude of the (FRF)14,15 and the form:

[= l/2&, Q = w l w d

Hence resonant frequency (wd) and the width(w) can be found from the magnitude diagramsusing the reference cursor. The value of thedamping factor [ was plotted against frequencyand the effect of coated lamina orientation andboundary conditions are studied and discussedas shown in Figs 5 and 6.

RESULTS AND DISCUSSION

The resonant frequencies, amplitudes anddamping factors of laminated coated beamshave been measured and analysed for differentlamina coating code numbers for two types ofcore (steel and aluminium).

The frequency spectra of core and coatedsteel beams are shown in Fig. 4; multi peakswere monitored within the selected range fre-quency.

Table 2 shows the variation of the funda-

mental frequency, amplitude and dampingfactor for different coated beams and for twocore materials, steel and aluminium, using twotypes of fixation. It can be indicated that theamplitude of the specimen coated by 0 fiberorientation is relatively low compared with theothers. This is due to the maximum stiffness atthis orientation. In contrast the amplitude ofthe specimen coated by 90 fiber orientation isrelatively high compared with the others. This isdue to the minimum equivalent stiffness at thisorientation.

Dual channelsignal analyzer

2034I I 1

Impact hammer8202

Fig. 3. Instrumentation set-up formed from excitation and measuring systems.


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Vibr at io n and dampi ng anal ysis of coat ed beams 37

Table 2. Values of fundamental frequency, amplitude and damping factor for different coated lamina experimentalresults)

Core

type

Steel

Alum.

Steel

Alum.

Boundarycondition

C-f

C-f

c-c

c-c

Freq.Ampl.Damp. Fat.Freq.Ampl.Damp. Fat.Freq.AmpI.Damp. Fat.Freq.AmpI.Damp. Fat.

Withoutcoating

144 Hz18 dBo-004

143 Hz25 dBo-15

896 Hz8 dB0.002

895 Hz12 dB0.02

0=0

160 Hz29 dB0.023

181 Hz42 dB0.25

995 Hz12 dB0.012

1122 Hz17 dB0.06

Coated lamina orientation

0=90 8=45

140 Hz 141 Hz45 dB 42 dB0.08 0.06

137 Hz 139 Hz66 dB 63 dB0.35 0.33

883 Hz 894 Hz21 dB 19 dB0.040 0.03

871 Hz 878 Hz34 dB 32 dBO-16 0.14

Random

147 Hz35 dB0.030

145 Hz53 dBO-29

904 Hz17 dB0.015

906 Hz26 dB0.12

Table 3. Values of first five frequencies for core and coated steel using numerical and experimental method for differentboundary condition

Mode Boundary Steel without Coated steelno condition coating

8=0 8=90 l3=45 RandomMethod Num EXP

Num EXP Num EXP Num EXP Num EXP

I C-F 140.6 144 156 160 138 140 139.3 141 141 147c-c 898.3 896 997.2 995 881 883 890.6 894 901 904

II C-F 884.4 889 981.7 986 874 879 876 880 878.5 881c-c 2487 2493 2762 2767 2460 2464 2467 2470 2471 2474

III C-F 2477 2481 2750 2757 2449 2453 2456 2460 2486 2490c-c 4858 4860 5392 5394 4802 4807 4815 4819 4875 4879

IV C-F 4856 4853 5390 5394 4800 4804 4814 4817 4873 4875c-c 7170 7176 7948 7953 7087 7091 7108 7112 7196 7199

V C-F 8028 8025 8911 8916 7936 7941 7958 7964 8056 8061c-c 12002 12007 13329 13333 11864 11870 11898 11896 12044 12047

where C=clamped; F=free.

Fig. 5 and Fig. 6, it is observed that the damp-ing factor is small for coated orientation with8=0 compared with the other code coat num-bers. This is explained by the fact that fiberorientations in these directions are expected to

increase the core beams stiffness. Also, forcoated orientations with 0=90, the dampingfactor is high compared with the other orienta-tions. This direction decreases the core beamstiffness, where the maximum energy dissipationresults in a large system damping factor.

Therefore, according to practical require-ment, the natural frequencies of metalliccomponents can be controlled by coating lami-nated layers of various code numbers.

For the sake of comparison, the measuredand computed values of frequencies for two

cases of fixations (C-F, C-C) are listed in Tables3 and 4.

The natural frequencies of the aluminiumcoated beams with 6=0 are higher than thoseof steel coated beams are indicated in Tables 3and 4, from which it can be seen that the high-est structural damping is for aluminium while

the lowest value of damping capacity was mon-itored for steel beams as shown in Figs 5 and 6.The comparison between the experimental

and numerical results of the frequencies indi-cates a good agreement.

CONCLUSIONS

In the present study, the dynamic analysis ofvarious lamina coated orientations of glass/polyester beams has been investigated

experimentally and verified numerically in termsof equivalent stiffness and mass parameters. For


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38 A. A. El-Humid Hamada

Table 4. Values of first five frequencies for core and coated aluminium using numerical and experimental method fordifferent boundary conditions

Mode Boundaryno condition

Method

Aluminiumwithoutcoating

Num EXP

Coated aluminium

0=0 t3=90 8=45 Random

Num EXP Num EXP Num EXP Num EXP

I C-F 139.7 143 175 181 135 137 136.9 139 141 145c-c 892.7 895 1120 1122 868 871 874 878 901.7 906

II C-F 878.8 882 1103 1107 840 843 861 865 887 889c-c 2473 2478 3104 3109 2364 2370 2422 2426 2496 2499

III C-F 2462 2467 3090 3094 2395 2399 2412 2417 2487 2490c-c 4827 4830 6060 6062 4696 4701 4730 4734 4875 4879

IV C-F 4825 4831 6057 6063 4695 4700 4728 4734 4874 4877c-c 7125 7132 8944 8948 6932 6937 6982 6990 7200 7207

C-F 7977 7982 10014 10019 7761 7765 7816 7821 8057 8061c-c 11927 11932 14947 14953 11603 11609 11686 11692 12046 12052

where C=clamped; F=free.

the sake of testing in the research laboratory,various specimens of different coated laminaorientation beams are efficiently fabricated byutilizing a hand layout technique. The resultsobtained from the present investigation showedthat the frequency response function (FRF) ofthe tested specimens is a good key for charac-terising the material response under variousdynamic conditions. From the present study onecan indicate that:

(1)

(2)

(3)

coated laminated beams provide highdamping capacities compared with iso-tropic single metal beams;the lamina coated code numbers have adominant effect on the quantitativenature of the dynamic characteristics ofthe structures;in contrast to the dynamic behaviour ofcomposite structures,16 the experimentaland numerical results of the coated sam-ples fabricated here, indicate thepossibilities to improve the dampingcapacities without considerable variationsin the natural frequencies of the structurewithin a certain frequency spectrum anda selected coated code number, (and viceversa) as shown in Table 2, Figs 5 and 6.The result is that the present techniqueprovides an efficient tool for the controlof dynamic structures.

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