IMPROVEMENT OF THE DAMPING PROPERTIES OF CARBON-FIBRE-REINFORCED LAMINATED PLASTICS USING DAMPING LAYERS
IZBOLJŠANJE DUŠENJA Z OGLJIKOVIMI VLAKNI OJAČANE LAMINIRANE PLASTIKE Z UPORABO PLASTI ZA DUŠENJE
Radek Kottner1, Josef Varik2, Robert Zem«k3
University of West Bohemia, European Centre of Excellence, New Technologies for Information Society, Univerzitni 22, 30614 Plzen,
Czech Republic
2University of West Bohemia, Department of Machine Design, Univerzitni 22, 30614 Plzen, Czech Republic 3University of West Bohemia, Department of Mechanics, Univerzitni 22, 30614 Plzen, Czech Republic
kottner@kme.zcu.cz
Prejem rokopisa — received: 2012-09-03; sprejem za objavo - accepted for publication: 2012-10-08
A suitable hybrid composite consisting of carbon-fibre-reinforced plastic and damping layers was investigated in terms of damping and natural frequencies using experiments and numerical simulations. The frequency response and the transient response of cantilever beams were analysed. The damping layers made from rubber or from a cork-rubber composite material were used in the investigated hybrid structure. A laser-measurement device and an accelerometer were used for the measurement of the responses. Pareto optimization was performed using three-dimensional numerical simulations with the aim to maximize the fundamental natural frequency and the damping ratio. Keywords: hybrid, composite, carbon-fibre-reinforced plastic, rubber, damping
Sestavljeni kompozit iz plastike, ojačane z ogljikovimi vlakni in s plastmi za dušenje naravnih frekvenc, je bil preiskovan eksperimentalno in z numerično simulacijo. Analiziran je bil frekvenčni odgovor in prehodni odziv konzolnega nosilca. V preiskovani hibridni strukturi je bila plast za dušenje izdelana iz gume ali sestavljena iz gume in plute. Za meritve odgovora je bil uporabljen laserski merilnik in merilnik pospeška. Z uporabo tridimenzionalne numerične simulacije je bila izvršena Paretova optimizacija za maksimiranje osnovne naravne frekvence in količnika dušenja.
Ključne besede: hibrid, kompozit, z ogljikovimi vlakni ojačana plastika, guma, dušenje
1	INTRODUCTION
Recently, the conventional metallic structures have been replaced with composite structures thanks to their superior dynamic characteristics. For example, a higher specific stiffness of carbon-fibre-reinforced plastics (CFRPs) allows higher natural frequencies of the structures. Furthermore, the damping characteristics of CFRPs have higher values and, moreover, can be improved using an integration of the damping layers1.
The aim of this work was to develop a suitable hybrid structure in terms of damping and natural frequencies. The investigated samples were the cantilever beams, whose damping layers were made from rubber or from the ACM87 cork-rubber composite material. The damping ratio for their fundamental natural frequency was analyzed.
2	THEORY
The equation of the motion of a damped system is2: Mq + Cq + Kq = f(t)	(1)
where q[m], q[m s-1], q[m s-2] are the vectors of the generalized coordinates and their 1st and 2nd differentiations with respect to time t; f(t)[N] is the vector of the
generalized time-dependent applied force; M[kg] is the mass matrix of the system; C[Nsm-1] is the damping matrix and i[Nm-1] is the stiffness matrix. The motion of the discrete linear systems with a single degree of freedom can be described as:
mq + cq + kq = F(t)	(2)
In the case of free oscillations, Equation (2) can be consequently rewritten as:
q + 2£©0 q+©J q = 0	(3)
T
At - <p0/ffi
Figure 1: Response of an underdamped harmonic oscillator Slika 1: Odgovor harmoničnega nedušenja
where the damping ratio £ and the undamped natural frequency «0 are defined as:
C
2^/mk
ton = ■
(4)
(5)
In the case of an underdamped system (0 < £ < 1) the solution of Equation (3) representing the displacement of the system can be found in the following form:
q(t) = Ce^' sin(«t+p0)	(6)
where C[m] is the amplitude, «[rads-1] is the damped natural frequency of the system and po[rad] is the phase shift.
The damped natural frequency can be expressed as:
I-2 2n
«=«0V 1-C = T 2nf
(7)
where T[s] is the period of the waveform (Figure 1) and f is the damped natural frequency in Hertz.
Based on Equation (6), the exponential attenuation rate is then defined as:
b = £«0	(8)
The damping ratio can be determined using the logarithmic decrement 5, which is defined as the natural logarithm of any two peaks:
2n£
d = ^ln — = lne^T = T = bT =- ,-
n qn	0 ^
(9)
/3 = -
w
(12)
When the difference between the undamped natural frequency «0 and the damped natural frequency « is negligible, the Rayleigh constants of the hybrid composite components can be determined as:
2£ (
/3 ( =	(13)
« 0,h
where £i is the damping ratio of the hybrid composite component (CFRP or the damping material) and «0,h is
the undamped natural frequency of the hybrid composite determined from the modal analysis.
3 EXPERIMENTS
Two types of CFRP and two types of damping layers were used in the experiments. A linear behaviour of all the materials was assumed. The mechanical properties of the materials are listed in Tables 1 to 4.3 A modal analysis was used for an identification of the elastic-material properties4.
Two cantilever flat bars consisting of unidirectional 913C-HTS CFRP and rubber were investigated. Sample A had a thickness of 7.8 mm (Figure 2a), sample B had a thickness of 12.9 mm (Figure 2b). The thickness of 913C-HTS CFRP was 2.7 mm and the thickness of rubber was 2.4 mm. The width of the bars was 19.8 mm, the length was 450 mm and the length of the clamping was 38 mm. The harmonic response after the initial deflection was investigated using an optoNCDT
where q0 is the greater of the two amplitudes and qn is the amplitude of a peak n periods away. The damping ratio is then found from the logarithmic decrement:
<5
£ = 1 2 ,2	(10)
V4^2-5
According to Rayleigh damping, the damping matrix C is given by:
C = aM + ¡iK	(11)
where a and i are the Rayleigh constants. Assuming a = 0 and the damping ratio £ of both the CFRP and the damping materials is constant, the Rayleigh constant can be expressed as:
2£
Figure 2: Cantilever flat bar consisting of 913C-HTS CFRP and rubber: a) sample A, b) sample B
Slika 2: Konzolna ploščata palica iz 913C-HTS CFRP in gume: a) vzorec A, b) vzorec B
c
Table 4: Mechanical properties of the ACM87 composite Tabela 4: Mehanske lastnosti kompozita ACM87
Figure 3: Cantilever square tube consisting of K63712 CFRP and the ACM87 cork-rubber composite material: sample C Slika 3: Konzolna kvadratna cev iz K63712 CFRP in ACM87 kom-pozitnega materiala pluta-guma; vzorec C
laser-measurement device or a Bruel & Kjaer 4507 accelerometer.
Table 1: Mechanical properties of 913C-HTS CFRP Tabela 1: Mehanske lastnosti 913C-HTS CFRP
Longitudinal modulus	Ei	GPa	104
Transverse modulus	E2	GPa	5.5
Shear modulus	Gl2	GPa	2.4
Poisson's ratio	V12	—	0.34
Density	P	kg/m3	1.500
Damping ratio	1	-	0.002
Table 2: Mechanical properties of K63712 CFRP Tabela 2: Mehanske lastnosti K63712 CFRP
Longitudinal modulus	E1	GPa	280
Transverse modulus	E2	GPa	3.5
Shear modulus	G12	GPa	1.7
Poisson's ratio	V12	-	0.38
Density	P	kg/m3	1.470
Damping ratio	1	-	0.003
Table 3: Mechanical properties of rubber Tabela 3: Mehanske lastnosti gume
Young's modulus	E	MPa	10
Poisson's ratio	V	-	0.49
Density	P	kg/m3	1 170
Damping ratio	1	-	0.072
Young's modulus	E	MPa	2.5
Poisson's ratio	V	-	0.3
Density	P	kg/m3	740
Damping ratio	1	-	0.112
Table 5: Results of the experiments and models Tabela 5: Rezultati eksperimentov in modelov
	Damped natural frequency f /Hz		Damping ratio 1	
	Exp.	Model	Exp.	Model
Sample A	35.5	37.8	0.034	0.031
Sample B	44.3	43.8	0.043	0.041
Sample C - clamped	44.0	43.7	0.039	0.004
Sample C - free	548	549	0.005	0.004
Figure 4: Boundary conditions of a model of a cantilever flat bar Slika 4: Robni pogoji modela konzolne ploščate palice
Figure 5: Model of a hybrid square tube Slika 5: Model hibridne kvadratne cevi
Figure 6: Pareto optimization results, "C" means CFRP, "D" means the damping layer
Slika 6: Rezultati Paretove optimizacije: "C" pomeni CFRP, "D" pomeni plast za dusenje
Further, the response of the cantilever square tube consisting of the wound K63712 CFRP and the ACM87 composite with a thickness of 6.5 mm (Figure 3) was investigated (Sample C). The thickness of K63712 CFRP was 2.4 resp. 2.1 mm and the thickness of the ACM87 composite was 2 mm. The width of the square tube was 103 mm, the tube length was 1 490 mm and the length of the clamping was 135 mm. However, the clamping was not sufficiently rigid. Therefore, the damping ratio was investigated also in the case of a free square tube (a tube hung on a rope).
The results are listed in Table 5.
4 NUMERICAL SIMULATIONS
Three-dimensional finite-element models were created in MSC.Marc using eight-node brick elements (with an assumed strain option) as shown in Figures 4 and 5. The CFRP materials were considered orthotropic and homogenous. The damping materials were considered isotropic and homogenous. The boundary conditions of the models of the cantilever beams are obvious from Figure 4. The undamped natural frequencies were obtained with a modal analysis. After an evaluation of the Rayleigh constant using Equation (12), the damped natural frequencies and the damping ratio were obtained with a transient analysis. The Newmark time integration scheme was used.
Figure 7: Detail of the Pareto front Slika 7: Detajl Paretove linije
The created models were validated by comparing the experimental and numerical results listed in Table 5. The difference between the results for samples A and B was less than 9 %. In the case of the clamped sample C, the clamping was not sufficiently rigid as already mentioned above. Due to this fact, the damping ratio had a significantly higher value, which was confirmed with the experiment with the free sample C. The difference between the results involving the free sample C was less than 20 %.
A Pareto optimization was performed for the cantilever flat bar with the aim to maximize the fundamental natural frequency and the damping ratio. The analyzed bar was a symmetrical hybrid laminate with 14 layers consisting of 913C-HTS CFRP and rubber. The thickness of the layers was 1 mm and the fibre orientation of the CFRP layers was identical to the beam axis. The width of the bar was 20 mm, the length 450 mm and the length of the clamping was 30 mm. The Pareto front is shown in Figure 6, a detail of the front in Figure 7. It is obvious from both figures that the situation is very complex; therefore, the problem should be more precisely constrained.
5 CONCLUSION
The performed experiments and numerical simulations showed that the investigated natural frequencies and damping ratios of the hybrid composite structures
strongly depend on the placement of the damping layers in the hybrid cross-sections. The accurate place of the damping layers must be investigated for each application in dependence on the material of the layers, the geometry and the requested ratio between the fundamental natural frequency and the damping ratio.
Acknowledgement
This work was supported by the European Regional Development Fund (ERDF), within the project "NTIS -New Technologies for Information Society", European Centre of Excellence, CZ.1.05/1.1.00/02.0090, and by the research project no. P101/11/0288.
6 REFERENCES
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