Static and dynamic behavior of Rubbercork Composite materials

Transcription

1 Static and dynamic behavior of Rubbercork Composite materials Ivo Machado Guelho Instituto Superior Técnico Portugal, October 2011 ABSTRACT The present thesis studies the static and the dynamic response of different types of rubbercork composite materials in order to characterize their mechanical behavior. This work presents a modal analysis based study to determine the materials dynamic behavior and their energy absorbing capacity. To accomplish these purposes, a measuring chain was specially designed to simulate the single degree of freedom modal system. Afterwards, experimental data were obtained by doing quasi-static compression tests in a servo-hydraulic machine and vibration tests in an electrodynamic shaker. The accuracy in determine the modal parameters can be particularly affected by the chosen method or even by the characteristics of the Frequency Response Functions used. Therefore, three methods were used. The well-known Half-Power bandwidth method used to determine natural frequency and damping ratio and two other methods based on the identification and extraction of modal parameters. They were the Circle-Fit method and the Fraction Rational Polynomial method. The main goal of this work was achieved when all the dynamic properties of the materials such as loss factor, natural frequency, elastic moduli (storage and loss modulus) and the complex dynamic stiffness K*(jω), among others, were determined and when analytical equations representing the dynamic behavior of the materials were established. 1 INTRODUCTION Composite materials are engineered materials from two or more materials with significantly different physical, chemical and mechanical properties. The rubbercork composite materials specifically started to be developed in the 1960 s and they were initially used in oil sealing applications in the automotive industry [1]. Nowadays, with a wider range of applications, in the acoustic, heat, electric and vibration isolation fields, the characterization of these composites becomes essential to get better performances in each application. Currently, one of the most important applications of these materials in engineering is as energy dissipation mechanisms, namely as passive damping devices. Thus, this thesis tries to characterize these composite materials and determine their vibration absorbing ability by determine their damping, by presenting a modal analysis study. The modal analysis is a strong engineering tool that appeared in the 1940 s and provided better comprehension about the dynamic properties of structures under dynamic excitation [2]. Extracting modal parameters like natural frequencies, loss factors, phase angles and 1

2 residues (inherent to the contribution of other modes) are essential in modal testing, since the experimental frequency response functions (FRFs) must fit with the curves obtained from the analytical equations. The accuracy of this extraction can be particularly affected by the chosen method or even by the characteristics of the Frequency Response Functions used [3]. Therefore, three methods were used. The wellknown Half-Power bandwidth method used to determine natural frequency and damping ratio and two other methods based on the identification and extraction of modal parameters. They were the Circle-Fit method and the Fraction Rational Polynomial method. This thesis pays particularly attention in understanding how efficient the dynamic behavior of these materials can be described by the simplest modal analysis system, the single degree of freedom system and how accurate the extraction techniques used in extracting modal parameters are. body; the damping is represented by a dashpot which is considered as having no stiffness or mass. a) b) Figure 2.1 Discretized representation of: a) SDoF system with viscous damping; b) SDoF system with hysteretic damping [4]. The application of the Second Newton s Law to the previous models and summing all the forces on the masses we get the following differential equations of motion for the two damping types: Viscous damping: (2.1) 2 THEORETICAL ANALYSIS 2.1 Vibration theory Hysteretic damping: (2.2) To accomplish the main objective in determine the dynamic properties inherent of the rubbercork materials, a strong engineering tool called Modal Analysis was used. Although Modal Analysis is largely based on the analysis of multi-degree-offreedom (MDoF) systems with an inherent matrix theory background, this work tries to simplify the characterization of these viscoelastic materials by using the simplest vibratory system, the singledegree-of-freedom (SDoF) system. This system is a simple discretization of a physical model into a spring-mass-damper system whose motion can be described by a single variable x (Figure 3.1). Each one of these three elements (spring-massdamper) represents a unique property of the system where: the spring represents the stiffness and is considered massless; the mass represents the inertia and is treated as an infinitely rigid If an harmonic force is applied to the system,, then the response will be also harmonic and will take the form of. Taking this into account, the equations (2.1) and (2.2) can now be written as (2.3) (2.4) Where is the frequency response function (FRF) of the system. The FRF is the main function on which modal analysis will depend and although in theory the FRF is dictated only by the system, in reality the accuracy of measured FRF data is critical to the success of modal analysis. Since this FRF uses the displacement as the 2

3 response, normally is denoted as receptance or inertance of the system. From the Receptance equations an important concept in vibrations theory called Transmissibility, which describes the effectiveness of vibration isolation system, can be deduced: (2.7) (2.8) The equation (3.1.19) can also be written as: (2.9) (2.5) By applying the Hooke s law (σ=eε) to equations (2.8) and (2.9) the Young modulus can also be written in two components: (2.10) (2.6) and (2.11) Viscoelasticity Viscoelastic materials are characterized by having an intermediate behavior between perfectly elastic and perfectly viscous behavior. They are typically polymers, polymers solutions, amorphous materials or metals at very high temperatures. Regarding that perfectly elastic materials present a in phase(δ=0º) stress-strain response while the perfectly viscous materials show an out phase(δ=90º) lag, the viscoelastic materials behavior is evidenced when the materials are subjected to an oscillatory strain with frequency, where the stress-strain sinusoidal response curves show a phase lag or phase angle (δ) between stress and strain. By representing the storage and loss moduli in the complex plane, the complex Young modulus E* can be defined as (2.12) The previous equation (2.12) permits to define an important parameter called the loss factor η, which represents the damping capacity of the material, described as (2.13) Half-Power Bandwidth Method Figure Phase lag between stress and strain [5]. The stress and strain equations from Figure 3.2 can be defined as: The half-power bandwidth method is a very good and simple technique to determine the damping ratio and therefore the loss factor of low-damped systems. Considering a SDoF system, the basic principle of this method consists in: first, to find the value of the quality factor, Q, which represents the peak magnitude of receptance α(ω), at the resonant frequency; second, locate the magnitude values where the magnitude is 3

4 1 2 times of the amplification factor and get the two frequencies ω1 and ω2 (Figure 3.8); finally, to create a band around the resonance frequency with those two frequencies. (2.14) Where represents the real modulus of gain with a phase angle. This angle symbolizes the rotation of the circle. The value is a complex constant representing the contibutions of other modes in the frequency range surrounding a single resonance peak [8]. Figure 2.3 Bandwidth method of damping measurement in a single-dof system [6] Circle-Fit Method The Circle-Fit method is no more than a curve fitting process which allows determining the analytical parameters for the frequency transfer function from the analysis of the Nyquist plot. In most cases, the circle is rotated and displaced from its place, as shown in the Figure This is explained by the fact that other vibration modes are also present despite its influence can be neglected for values around the resonance frequency. In Maia [4] and Ewins [7] the loss factor is derived and by trigonometry relations it was found to be (2.15) It is also known that the best results of η r should be obtained when the angles take similar values and when they are not too small (not too close the resonance frequency). In line with what was said and regarding equation (2.14) the circle diameter is given by: (2.16) This can be seen in Figure 2.4 where the phase angle is also represented as being as (2.17) Figure 2.4 Circle-Fit example with r vibration modes The coordinates represent the center of the circle and the coordinates are the displaced values from the origin. This coordinates relative to the translation of the circle are given right after the resonance frequency being located (Figure 2.4). In Ewins [7] the receptance curve for the Circle-fit approach is described as being: 4

5 2.1.4 Rational Fraction Polynomials method (RFP) The process of identifying parameters from an FRF function is commonly called curve fitting or parameter estimation. Rational Fraction Polynomial method (RFP method), performs curve fitting on the measurement (FRFs) curves in order to identify the modal parameters (natural frequencies, damping ratios, loss factors, etc.) of the predominant modes of vibration of the structure, and can also be used to identifying poles, zeros and resonances of combined electro-mechanicalacoustic systems [9]. By assuming that the frequency response measurements are taken from a linear, second order dynamical system, it can be represented by the ratio of two polynomials. resonant systems, the FRF analytical curve can be represented in a partial fraction form. This form describes the FRF in terms of parameters, giving for n-degrees of freedom n-pole pairs and clearly differentiates the residues r k associated to each pole pair [9]. Partial Fraction Form (2.19) (2.20) After gathering all the polynomial coefficients, the natural frequencies ω n of the p k pole are [10]: (2.21) where the real part of the pole, from [4] and [9], is equal to (2.22) It is now time to rewrite equation (2.19) in order to damping ratio. By substituting equation (2.21) in equation (2.22), the damping ratio becomes, (2.23) Figure 2.5 Curve fitting using Rational Fraction Polynomials. By curve fitting (Figure 3.16) the experimental curve, the analytical equation (3.1.47) with the polynomials coefficients can be determined in a least squared error criterion. Modeling the formulated problem in Laplace domain derives in Rational Fraction Form (2.18) This is called the Rational Fraction Form and is only useful when poles are located along the damping axis (non-resonant systems). For Since that this model was established for viscous damping models, we should use the equation η 2ζ for lightly damped materials Modal fitting curve analysis In order to help validate the loss factor values when one of the methods fails, iterations were made by using the receptance equation (2.4) for a hysteretic model. These iterations were made by making a substituting the loss factor values in the referred equation that could better approximate the analytic curve to the real one. The fit was made using the Least squares method. It consists in adjusting the parameters of a model function to best fit a data set. A simple data set consists of n points (data pairs), i=1,,n, where is an independent variable and 5

6 is a dependent variable whose value is found from the real curve. The analytical function takes the shape of f(x,β), where the m adjustable parameters are held in the vector β. In this case the varying parameter is the loss factor η. The goal is to find the parameter values for the model which "best" fits the data. The least squares method finds its optimum when the sum, S, of squared residuals is a minimum. bounded between two Ck45 steel disk plates with the same diameter. The Ck45 steel has a very high Young modulus ( GPa) and therefore a high stiffness value. The choice of this material for the disks was particularly important because it should prevent steel resonance from being nearby the frequency range tests and should also guarantee significant stiffness that shall prevent bending and assure that won t be any extra vibration mode. (2.24) The difference between the actual value of the dependent variable and the value predicted by the model is called residue, (2.25) a) b) c) Figure 3.1 Measuring chain a) Ck45 steel disk plate after faced in lathe; b) Rubbercork specimen; c) Final measuring chain ready for tests. 3 MATERIALS AND EXPERIMENTAL PROCEDURE 3.1 Materials All the materials involved in this work were provided by the Amorim Cork Composites, the world leading company in the production of cork. The materials had the commercial designations: VC1001, VC2100, VC6400, VC5200, polychloroprene rubber (neoprene) and NL20 (natural cork). 3.2 Experimental Procedure In order to characterize the rubbercork aglomerates, models that could better describe the Single Degree of Freedom modal system started to be constructed for the vibrations tests taking into account the ASTM and the ISO standards, which were the main standards found that could be helpful in this work. 3.4 Quasi-static tests To characterize the mechanical behavior of these materials, primarily quasi-static compressive experiments were carry out to determine the Young s modulus, among other parameters, from the experimental stress-strain curves. The quasi-static uniaxial compressive tests were performed in an electro-mechanical testing machine (INSTRON 5566) available at the Mechanical Laboratory with a 50kN load cell capacity. The testes carried out at room temperature of 23ºC with a fixed loading speed of 1mm/min to ensure a quasi-static response. 3.3 Specimens construction Considering the standards, and regarding the materials thickness provided by Amorim, all the rubbercork specimens (29 specimens) were cut in disk plates with 80 mm of diameter and later 6 Figure Photograph of the quasistatic compression tests.

7 Nominal stress [MPa] Nominal stress [MPa] These quasi-static tests intended to be illustrative of the three different regions of the stress-strain curves relative to viscoelastic materials and how different the static behavior is in different rubbercork composites. Another purpose is to show how shape factor affects the stress-strain curves. 3.5 Vibration tests The vibration tests were performed in the Laboratory of Vibrations also in the Mechanical Department in Instituto Superior Técnico Experimental setup The setup taken in this work in order to gather the FRF curves and characterize the dynamic properties of the materials is presented in Figure 3.3. It consists in a shaker which will provide an external force to the system. This force will be measured by the force transducer and for the accelerometer 1, which will give the input force and the acceleration of the lower disk, respectively. Then, an accelerometer 2 will gather the acceleration in the superior disk and also gives to the data acquisition and posteriorly analyzed by analysis software. Figure 3.3 Experimental setup. All the tests were performed at room temperature of 23ºC. The tests were accomplished by transmitting a random sine wave with the shaker to the bottom of the Ck45 steel plate, as shown in Figure RESULTS 4.1 Static Behavior The compression test performed to the VC1001 (10 mm thickness) show that this is a very elastic material behaving almost linearly elastic till 40% of compressive strain, see Figure 4.1. It also shows two different regions are well identified: an almost plateau region till 40% of strain and the densification region from 60-80% of strain. 10,0 8,0 6,0 4,0 2,0 0,0 0 0,2 0,4 0,6 0,8 Nominal strain [mm/mm] Figure Compression test in a VC1001 specimen with 10mm of thickness. However, amplifying the first 10% of strain region, it can be seen that the material presents linear elastic behavior. Plotting a tendency line till 8,5% of strain gives an elastic modulus of 0,71 MPa with a stress-strain correlation of 99,92% (Figure 4.2). 0,06 0,05 0,04 0,03 0,02 0,01 y = 0,7106x - 0,0034 R² = 0,9992 0,00 0,00 0,02 0,04 0,06 0,08 0,10 Nominal strain [mm/mm] Figure 4.2 Amplification of the linear elastic region from Figure

8 Magnitude [db] At the end of the test, the elastic recovery of the material was measured, and after 1 minute of recovery the result was a 98%. 4.2 Dynamic Behavior The accelerance curve of the VC1001 material (Error! Reference source not found. Error! Reference source not found.) acquired by the application of a force to the measuring chain, as well as the mobility and receptance curves analytical determined, are presented in Figure Accelerance Mobility Receptance Figure Nyquist circle-fit plot for VC test specimen Frequency [Hz] Figure Accelerance, mobility and receptance of VC1001 with 10mm of thickness. The implementation of the circle-fit started determining the corrected value of the natural frequency of the system through a finite difference table of the phase angle values. Then, the circle-fit method was implemented in Matlab, resulting in the following Figure 4.4 from which the loss factor values for different angles could be determined, see Figure 4.5 and Figure 4.6. Figure 4.5 Loss factor for different data points (3d view). 8

9 Figure Loss factor for different data points (top view). It can be seen from Figure 4.6 that a reasonable value of the loss factor can be chosen from a 45 degrees direction where the gradient of η is minimum. After determine the values of η from the Nyquist plot, all the modal parameters were able to be determined, allowing the determination of the Circle-fit receptance curve from equation (2.14) as well as the analytic receptance curve from modal analysis - equation (2.4). Consequently, the Figure 4.7 and the Figure 4.8 show the experimental and the numerical curves. Figure 4.8 Experimental (blue line) and numerical FRF curves (red line) of VC1001 material through analytical interpolation. All these tests and methods were used in all the materials however, only the VC1001 results are presented in this article. The Table 4.1 and Table 4.2 present all the dynamic properties for the VC1001 test specimens. Spec. L[mm] , , , , , ,178 [N/m] [MPa] [MPa] 1,50x10 6 6,00 1,17 1,57x10 6 6,24 1,11 1,25x10 6 4,95 1,01 1,21x10 6 4,80 0,87 1,69x10 6 6,72 1,25 1,89x10 6 3,77 0, ,187 3,31x10 6 6,60 1, ,215 1,78x10 6 3,54 0,76 Table 4.1 VC1001 dynamic properties determined from the Circle-fit method. Figure 4.7 Experimental (blue line) and numerical FRF curves (red line) of VC1001 material through the Circle-fit method Concerning the RFP method, all the modal parameters estimated have resulted in a characteristic polynomial potted in 9

10 η 4.3 Loss factor frequency dependence Finally, for the VC1001 and for the VC5200 materials were performed additional vibration tests by coupling more mass to the measuring chain representative of the SDoF system. The Figure 4.10 shows two of the four tests made in the vibrations laboratory. By picking randomly two specimens of different materials (VC specimen 5 and VC specimen 11) and adding consecutively Ck45 disk plates, vibration tests were performed with the double and the triple of the initial mass of the system, see Figure Figure 4.9 Experimental and RFP numerical curve of VC1001. Specimen [MPa] [MPa] 1 0, , , , , , , ,348 6,02 1,06 6,22 1,20 4,96 1,00 4,79 0,94 6,70 1,35 3,63 1,21 6,38 2,10 3,42 1,19 Table VC1001 dynamic properties determined from the RFP method. Comparing the Table 4.1 and the Table 4.2 values, it can be realized that both methods produce similar results for the first 5 specimens, where the thickness is equal to 20 mm. For specimens 6, 7 and 8 the RFP method takes unrealistic values of η when comparing with the other ones. This can be explained by the noise near the natural frequency and also by the noncorrected value of the natural frequency. a) b) Figure 4.10 a) Vibration test in a VC1001 specimen with three times the initial mass; b) Vibration test in a VC5200 specimen with two times the initial mass. 0,25 0,2 0,15 0,1 0, frequency [Hz] Figure 4.11 Loss factor frequency dependence in a VC1001 material. 5 CONCLUSIONS Concerning the methods used, it can be said that the Circle-fit method was the most reliable implemented method. The Half-Power method as expected is only useful for very lightly damped m 2m 3m 10

11 systems, were the peak is well-defined. Regarding the Circle-fit method, it reveals problems when the FRF curves show problems precisely around the resonance frequency, just like Imregun exposed in [11]. The RFP method turns out to be very sensitive to any existent noise and it is always necessary to select the right points around the resonance in order to obtain good results. Finally, fitting the way using the hysteretic model equation was in most cases correct and useful to validate the results from the other methods. 6 REFERENCES 1. Gil, Luis. Cortiça: Produção, Tecnologia e Aplicação. Lisboa : Instituto Nacional de Engenharia e Tecnologia Industrial, He, Jimin and Fu, Zhi-Fang. Modal Analysis. Delhi : Replika Press Pvt Ltd, Kennedy, C. C. and Pancu, C. D. P. Use of Vectors in Vibration Measurements and Analysis. s.l. : Journal of the Aeronautical Sciences, November Maia, Nuno M. M. and Silva, Julio M. M. Theoretical and Experimental Modal Analysis. Lisbon : Research Studies Press Ltd, Riande, Evaristo, et al., et al. Polymer Viscoelasticity. New York : Marcel Dekker, Inc., Silva, Clarence W. de. Vibration and Shock Handbook. Vancouver : Taylor & Francis Group, LLC, measurements using rational fraction polynomials. San Jose, California : s.n., Policarpo, H., Neves, M. M. and Maia, N. M. M. On the determination of storage and loss moduli for cork composition materials. Lisbon : s.n., Imregun, M. A comparison of SDoF and Global MDoF Modal Analysis Techniques when applied to a Lightly-Damped Structure". s.l. : Proceedings of the International Modal Analysis Conference, Gil, L. and Silva, P. P. Cork Composites. Brighton : In ECCM9-Composites: From fundamentals to Exploitation, Snowdon, J. C. Vibration and Shock in Damped Mechanical Systems. Pennsylvania : John Wiley & Sons, Inc., NIELSEN and FUGLSANG, L. Material properties determined by vibration analysis. s.l. : Danish Society for Materials Testing and Research, Avalle, M., Belingardi, G. and Montanini, R. Characterization of polymeric structural foams under compressive impact loading by means of energy-absorption diagram. Sant' Agata - Messina : s.n. 16. Roylance, David. [Online] October engineering/3-11-mechanics-of-materials-fall- 1999/modules/visco.pdf. 7. Ewins, D. J. Modal Testing: Theory and Practice. London : John Wiley & Sons Inc., Briggs, John Charles. Force Identification using Extracted Modal Parameters, with Applications to Glide Height Testing of Computer Hard Disks. Cambridge, Massachusetts : s.n., Richardson, Mark H. and Formenti, David L. Parameter estimation from frequency response 11