Advances in Vibration Analysis Research Part 5 docx

Free Vibration of Smart Circular Thin FGM Plate 109 To solve Eq. (34) for w, we first assume that; ( ) 1 (, ,) () im t wr t w re θ ω θ − = (34) where 1 ()wr is the displacement amplitude in the z - direction as a function of radial displacement only; ω is the natural angular frequency of the compound plate; and m is the wave number in the circumferential direction. Rewriting Eq. (32) in terms of 1 ()wr and using Eq. (34), after canceling the exponential term one would get; 22 31211101 0PwPw Pw Pw ΔΔΔ ΔΔ ω Δ ω − −+= (35) where 22 22 ddr drdrmr Δ =+− Eq. (35) can be solved by the method of decomposition operator and noting that the 1 w is non-singular at the center of the plate its general solution yields to 3 1 1 () nm nm n n wAZr α = = ∑ (36) here 11 x α = , 22 x α = , 33 x α = (37) in which x 1 , x 2 and x 3 are the roots of the following cubic characteristic equation, 322 2 32 1 0 0Px Px Px P ωω − −+= (38) and (), 0 () (,) (), 0 mi i im i im i mi i Jrx ZrZ r Irx α αα α < ⎧ == ⎨ > ⎩ (39) here i=(1,2,3) and () mi Jr α , () mi Ir α are the first type and the modified first type Bessel function ,both of them of the order of m. In order to obtain appropriate solution for (, ,)rt ϕ θ , we assume; ( ) 1 (, ,) () im t rt re θ ω ϕθ ϕ − = (40) then substituting Eq. (36) in to Eq. (31)we arrive to the following relation for (, ,)rt ϕ θ ; 3 1 22 13133 31 1 42 12 110 11 () 16 (2 () )() nm p n n p n nnmn re Ahshe DD P Z r ϕΞ α αΞ ωΞ α − = ⎡ ⎡⎤ = − ⎣⎦ ⎣ ⎤ ++× ⎦ ∑ (41) 7. Case studies, results and discussions We will solve above the relations in this section; the material parameters and geometries are listed in Table 1. Advances in Vibration Analysis Research 110 FGM Plate: E c = 205 GPa ρ c =8900 (kg/ m 3 ) E m = 200 GPa ρ m =7800 PZT4: 11 E C = 132 33 E C = 115 55 E C = 26 GPa 13 E C = 73 12 E C = 71 e 31 =-4.1 (C/m 2 ) e 33 =14.1 e 15 =10.5 11 Ξ =7.124 (nF/m) 33 Ξ =5.841 ρ p =7500 (kg/ m 3 ) Geometry(mm): r 0 =600 h f =2, h p =10 Table 1. Material properties and geometric size of the piezoelectric coupled FGM plate [13,17] 7.1 Clamped circular piezo-coupled FGM plate The boundary condition is given by 11 1 0 0( )wdwdrddr atrr ϕ === = (42) and the characteristic equation is 11 12 13 21 22 23 31 32 33 0 ccc ccc ccc = (43) 10200 23 5 4 0 1 2 0 11 1 2 11 3 0 223 31 31 0 (), () () () () 8 16 16 iimi iiimi pii p i pi i im i cZ r c rZ r hrs D Dhr D Dh cZr eer ααα ααΞαλΞ α ′ == ⎛⎞ ++ ⎜⎟ ′ =− + ⎜⎟ ⎝⎠ (44) 1 2 4 0 12 2( ) ff pp hh r DD ρρω λ ⎡ ⎤ + ⎢ ⎥ = + ⎢ ⎥ ⎣ ⎦  (45) 2 12 2 0 2( ) ff pp DD hh r λ ω ρρ + = +  (46) in which the ()’ symbol indicates the derivative with respect to r and λ is the nondimensional angular natural frequency. After calculating ω from Eq. (43) and using Eqs. (36, 42) we find the mode shape for w 1 as; 3 2 20 3 30 2 3 30 2 20 13 11 2110220 1220110 1330110 3110330 21 10 2 20 12 20 1 1 ()() ()() () ( ) ()() ()() ()() ()() ()() ()( mm mm m m mm mm mm mm mm mm ZrZr ZrZr wr A Z r ZrZr ZrZr ZrZr ZrZr ZrZr ZrZr αα ααα α α αα ααα α αα ααα α αα ααα α ⎡ ⎛⎞ ′′ − =× × + ⎢ ⎜⎟ ′′ − ⎢ ⎝⎠ ⎣ ′′ − ′′ − ] 22 33 0 () () ) mm ZrZr αα ⎛⎞ ×+ ⎜⎟ ⎝⎠ (47) and moreover, by using Eqs. (36, 41, 42) we have the electric potential as; Free Vibration of Smart Circular Thin FGM Plate 111 3220330 2330220 3 2110220 1220110 1 22 4 2 1 1 1 1 31 1 2 1 11 0 11 31 33 13 30 1 ()() ()() ˆ () ()() ()() ( ) (2 ( ) ) 16 () mm mm m mm mm mpp mm ZrZr ZrZr rA ZrZr ZrZr ZrhsheDD P e ZrZ αα ααα α ϕ αα ααα α αα αΞωΞΞ αα − ⎡ ⎛⎞ ′′ − =× × ⎢ ⎜⎟ ′′ − ⎢ ⎝⎠ ⎣ ⎡⎤ ⎡⎤ ×× −+ + × + ⎣⎦ ⎣⎦ ′ + 10 3 1 10 3 30 2110220 1220110 1 22 4 2 2 2 2 2 31 1 2 2 11 0 11 31 33 22 4 2 3 3 31 1 2 3 11 0 11 () ()() ()() ()() ( ) (2 ( ) ) 16 (2 ( ) ) 16 mm mm mm mpp pp rZrZr ZrZr ZrZr ZrhsheDD P e hs he DD P ααα α αα ααα α αα αΞωΞΞ ααΞωΞ − ⎛⎞ ′ − × ⎜⎟ ′′ − ⎝⎠ ⎡⎤ ⎡⎤ ×× −+ + × + ⎣⎦ ⎣⎦ ⎡⎤ +−++× ⎣⎦ 1 31 33 3 3 () m eZr Ξα − ⎤ ⎡⎤ ⎣⎦ ⎥ ⎦ (48) Power Index Mode no. FGM plate g m Present Method Present (FEM) Error (%) Wang et al. [13] 0 138.42 139.27 0.61 138.48 1 288.05 289.70 0.57 288.20 0 2 472.55 473.45 0.19 472.79 0 134.63 135.43 0.59 - 1 280.17 281.78 0.57 - 1 2 459.62 460.45 0.18 - 0 132.70 133.63 0.69 - 1 276.19 278.04 0.67 - 3 2 453.09 454.34 0.28 - 0 132.12 133.06 0.70 - 1 274.96 276.85 0.68 - 5 2 451.06 452.39 0.29 - 0 131.85 132.78 0.70 - 1 274.39 276.25 0.67 - 7 2 450.13 451.46 0.29 - 0 131.69 132.70 0.76 - 1 274.07 276.09 0.73 - 9 2 449.60 450.84 0.28 - 0 131.64 132.55 0.68 - 1 273.96 275.79 0.67 - 10 2 449.42 450.66 0.28 - Table 2. First three resonance frequencies (Hz) of FGM plate In order to validate the obtained results, we compared our results with those given in the literature [7,9,10].Further as there were no published results for the compound piezoelectric FGM plate, we verify the validity of obtained results with those of FEM results. Advances in Vibration Analysis Research 112 Our FEM model for piezo- FG plate comprises: a 3D 8-noded solid element with: number of total nodes 26950, number of total element 24276, 3 DOF per node in the host plate element and 6 DOF per node in the piezoelectric element. Tables 2 and 3 shows the numerical results of our method compared with other references and techniques. As one can see from Table 2, the obtained results from the analytical method when g=0 (isotropic steel plate) corresponds closely with the results of [7-9] and FEM solution. As it is seen in these tables the maximum estimated error of our solution with FEM is about 1.51%. Power Index Mode no. Coupled Piezo-FGM plate g m Present Method Present (FEM) Error (%) Wang et al. [13] 0 143.63 144.69 0.73 143.71 1 298.92 300.49 0.52 299.070 2 490.37 492.62 0.46 490.62 0 140.26 142.22 1.38 - 1 291.89 295.82 1.33 - 1 2 478.84 482.09 0.67 - 0 138.54 140.60 1.46 - 1 288.33 292.47 1.42 - 3 2 472.99 476.61 0.76 - 0 138.01 140.07 1.47 - 1 287.21 291.39 1.43 - 5 2 471.16 474.81 0.77 - 0 137.76 139.82 1.47 - 1 286.69 290.83 1.43 - 7 2 470.30 473.95 0.77 - 0 137.62 139.73 1.51 - 1 286.40 290.54 1.43 - 9 2 469.83 473.16 0.70 - 0 137.57 139.61 1.46 - 1 286.30 290.41 1.42 - 10 2 469.66 473.26 0.76 - Table 3. First three resonance frequencies (Hz) for piezo-coupled FGM plate for various values of power index A close inspection of results listed in Tables 2 and 3 indicates that the amount of error between analytical and FEM results for the natural frequencies in FGM plate alone in the all vibration modes and for all values of g are less than the similar results for the compound plate. The obtained results in Table 3 indicate that by increasing the value of g, the frequency of system decreases in all different vibrational modes. Moreover, this decreasing trend of frequency for smaller values of g is more pronounced, for example by increasing value of g from 1 to 3 (~200%) the frequency of the first mode for the compound plate decreases by Free Vibration of Smart Circular Thin FGM Plate 113 130 132 134 136 138 140 142 144 146 0246810 Power Index(g) Natural Frequency (Hz) FGM Plate-Analytical FGM Plate-FEM Piezo coupled FGM-Analytical Piezo coupled FGM-FEM Fig. 2. Effect of power index on the natural frequencies (first mode) 445 450 455 460 465 470 475 480 485 490 495 0246810 Power Index(g) Natural Frequency (Hz) FGM Plate-Analytical FGM Plate-FEM Piezo coupled FGM-Analytical Piezo coupled FGM-FEM Fig. 3. Effect of power index on the natural frequencies (third mode) 1.23% but by increasing g from 3 to 9 (~ 200%) of the same plate and for the same mode, the frequency decreases by 0.66%. In order to see better the effect of g variations on the natural frequencies of the different plates, Fig. 2 and Fig. 3 also illustrate these variations for the first and third mode shapes. As it is seen from Figs. 2 and Fig. 3, the behavior of the system follows the same trend in all different cases, i.e. the natural frequencies of the system decrease by increasing of g and stabilizes for g values greater than 7. In fact for g>>1 the FGM plate becomes a ceramic plate and the compound plate transforms to a laminated plate with ceramic core as a host plate. Advances in Vibration Analysis Research 114 8. Conclusion In this paper free vibration of a thin FGM plus piezoelectric laminated circular plate based on CPT is investigated. The properties of FG material changes according to the Reddy’s model in direction of thickness of the plate and distribution of electric potential in the piezoelectric layers follows a quadratic function in short circuited form. The validity of the obtained results was done by crossed checking with other references as well as by obtained results from FEM solutions. It is further shown that for vibrating circular compound plates with specified dimensions, one can select a specific piezo-FGM plate which can fulfill the designated natural frequency and indeed this subject has many industrial applications. 9. References [1] Koizumi M. concept of FGM. Ceram. Trans. 1993; 34: 3–10. [2] Bailey T, Hubbard JE. Distributed piezoelectric polymer active vibration control of a cantilever beam. J. Guidance, Control Dyn. 1985; 8: 605-11. [3] Millerand SE, Hubbard JE. Observability of a Bernoulli–Euler beam using PVF2 as a distributed sensor. MIT Draper Laboratory Report, 1987. [4] Peng F, Ng A, Hu YR. Actuator placement optimization and adaptive vibration control of plate smart structures. J. Intell. Mater. Syst. Struct. 2005; 16: 263–71. [5] Ootao Y, Tanigawa Y. Three-dimensional transient piezo-thermo-elasticity in functionally graded rectangular plate bonded to a piezoelectric plate. Int. J. Solids Struct. 2000; 37: 4377–401. [6] Reddy JN, Cheng ZQ. Three-dimensional solutions of smart functionally graded plates. ASME J. Appl. Mech 2001; 68: 234–41. [7] Wang BL, Noda N. Design of smart functionally graded thermo-piezoelectric composite structure. Smart Mater. Struct. 2001; 10: 189–93. [8] He XQ, Ng TY, Sivashanker S, Liew KM. Active control of FGM plates with integrated piezoelectric sensors and actuators. Int. J. Solids Struct.2001; 38: 1641–55. [9] Yang J, Kitipornchai S, Liew KM. Non-linear analysis of thermo-electro-mechanical behavior of shear deformable FGM plates with piezoelectric actuators. Int. J. Numer. Methods Eng. 2004; 59:1605–32. [10] Huang XL, Shen HS. Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. J. Sound Vib. 2006; 289: 25–53. [11] Reddy JN, Praveen GN. Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plate. Int. J. Solids Struct 1998; 35: 4457-76. [12] Wetherhold RC, Wang S. The use of FGM to eliminate or thermal deformation. Composite Sci Tech 1996; 56: 1099-104. [13] Wang Q, Quek ST, Liu X. Analysis of piezoelectric coupled circular plate. Smart Mater. Struct 2001; 10: 229-39. [14] Reddy J.N, Theory and analysis of elastic plates, Philadelphia: Taylor and Francis, 1999. [15] Halliday D, Resniek R. Physics, John Wiley and Sons, 1978. [16] Brush DO, Almroth BO. Buckling of bars plates and shells. New York: Mac-hill, 1975. [17] Loy CT, Lam KL, Reddy JN. Vibration of functionally graded cylindrical shells. Int. J Mech Sciences 1999;41: 309-24 7 An Atomistic-based Spring-mass Finite Element Approach for Vibration Analysis of Carbon Nanotube Mass Detectors S.K. Georgantzinos and N.K. Anifantis Machine Design Laboratory Mechanical and Aeronautics Engineering Department University of Patras, GR 26500 Greece 1. Introduction Since their discovery in 1991 by Ijima [1], carbon nanotubes (CNT) have received much attention as a new class of nanomaterials revealing a significant potential for use in a diverse range of novel and evolving applications. Much of the interest in CNTs has focused on their particular molecular structures and their unique electronic and mechanical properties. For example, their elastic stiffness is comparable to that of diamond (1000 GPa), while their strength is ten times larger (yield strength 100 GPa). Furthermore, CNTs conduct heat and electricity along their length with very little resistance, and therefore they act as tiny electrical wires or paths for the rapid diffusion of heat. As a result, progressive research activities regarding CNTs have been ongoing in recent years. For more detail on theoretical predictions and experimental measurements of both mechanical and physical properties, see the comprehensive reviews in [2,3]. The combination of an extremely high stiffness and light weight in CNTs results in vibration frequencies on the order of GHz and THz. There is a wide range of applications in which the vibrational characteristics of CNTs are significant. In applications such as oscillators, charge detectors, field emission devices, vibration sensors, and electromechanical resonators, oscillation frequencies are key properties. An representative application is the development of sensors for gaseous molecules, which play significant roles in environmental monitoring, chemical process control, and biomedical applications. Mechanical resonators are widely used as inertial balances to detect small quantities of adsorbed mass through shifts in oscillation frequency. Recently, advances in lithography and materials synthesis have enabled the fabrication of nanoscale mechanical resonators that utilize CNTs [4,5]. The use of a CNT to make the lightest inertial balance ever is essentially a target to make a nanoscale mass spectrometer of ultrahigh resolution. Building such a mass spectrometer that is able to make measurements with atomic mass sensitivity is one of the main goals in the burgeoning field of nanomechanics. An inertial balance relies only on the mass and does not, therefore, require the ionization or acceleration stages that might damage the molecules being Advances in Vibration Analysis Research 116 measured. This means that a nanoscale inertial balance would be able to measure the mass of macromolecules that might be too fragile to be measured by conventional instruments [5]. Several efforts for the building of CNT-based sensors have been presented in the literature. Mateiu et al. [6] described an approach for building a mass sensor based on multi-walled CNTs with an atomic force microscope. Chiu et al. [7] demonstrated atomic-scale mass sensing using doubly clamped, suspended CNT resonators in which their single-electron transistor properties allowed the self-detection of nanotube vibration. They used the detection of shifts in the resonance frequency of the nanotubes to sense and determine the inertial mass of atoms as well as the mass of the nanotube itself. Commonly, multi-walled CNTs are less sensitive than single-walled CNTs. However, multi-walled carbon nanotubes are easier to manipulate and more economical to be produced, since they are both longer and have larger diameters than single-walled CNTs [8]. Hence, it is important to develop accurate theoretical models for evaluation of natural frequencies and mode shapes of CNTs. An excellent review article was recently published by Gibson et al. [9] that presents related scientific efforts in dealing with the vibrational behavior of CNTs and their composites, including both theoretical and experimental studies. Controlled experiments performed at nanoscale dimensions remain both difficult and expensive. Despite of this fact, Garcia-Sanchez et al. [10] have recently presented a mechanical method for detecting CNT resonator vibrations using a novel scanning force microscopy method. The comparison between experimental and theoretical methods pre- require the complete definition of all parameters such as the length of the vibrating nanotube, the nanotube type and other conditions that influence the vibrational behavior such as the slack phenomenon, nature of the support condition, environmental conditions and other influences. In an attempt to approach the vibration behavior of CNTs, various theoretical methods have been reported in literature. Molecular dynamics (MD) and molecular mechanics, as well as elastic continuum mechanics, are considered efficient because they can accurately and cost- effectively produce results that closely approximate the behavior of CNTs. . Each of the previously mentioned approaches offers different advantages, but also certain drawbacks. MD is an accurate method capable of simulating the full mechanical CNT performance. However, it carries a high computational cost that may be prohibitive for large-scale problems, especially in the context of vibration analysis. Molecular mechanics-based techniques, such as those in [11-13], have been used for vibration analysis of CNTs and have been shown to be accurate and also more computationally cost-effective than MD. Nevertheless, under such approaches, the modeling of atomic interactions requires special attention because the mechanical equivalent used to simulate the carbon-carbon bond deformations must be efficient for the studied problem. Generally, typical elements of classical mechanics, such as rods [14], beams [15, 16], springs [17-19] and cells [20] have been proposed including appropriate stiffness parameters, thus their strain energies are equivalent to the potential energies of each interatomic interaction. Furthemore, elastic continuum mechanics methods based on well-known beam theories have also been successfully used to evaluate the vibration characteristics of CNTs under typical boundary conditions [21-24]. Xu et al. [25] studied the free vibration of double-walled CNTs modeled as two individual beams interacting with each other taking van der Waals forces into account and supported with different boundary conditions between the inner and outer tubes. These methods have the lowest computational cost; however, they can compute only a subset (mainly the bending modes) of the vibrational modes and natural frequencies. An Atomistic-based Spring-mass Finite Element Approach for Vibration Analysis of Carbon Nanotube Mass Detectors 117 In terms of CNT mass detector function, the principle of mass detection using CNT-based resonators is based on the fact that the vibrational behavior of the resonator is sensitive to changes in its mass due to attached particles. The change of the resonator mass due to an added mass causes frequency shifts. The key challenge in mass detection is quantifying the changes in the resonant frequencies due to added masses. Based on this principle, the usage of computational tools, as presented in prevous paragraph, capable of simulating the vibrational behavior of CNT-based mass detectors is important for two reasons. First, they can cost-effectively predict the mass sensing characteristics of different nanoresonator types, thereby allowing the optimal design of detectors with a specific sensing range. Second, their cooperation with experimental measurements can improve the detection abilities of the nanodevice. With respect to theoretical studies on CNT-based sensors, Li and Chou [26] examined the potential of nanobalances using individual single-walled CNTs in a cantilevered or bridged configuration. Wu et al. [27] explored the resonant frequency shift of a fixed-free single-walled CNT caused by the addition of a nanoscale particle to the beam tip. This was done to explore the suitability of a single-walled CNT as a mass detector device in a micro-scale situation via a continuum mechanics-based finite element method simulation using a beam-bending model. Chowdhury et al. [28] examined the potential of single-walled CNTs as biosensors using a continuum mechanics-based approach and derived a closed-form expression to calculate the mass of biological objects from the frequency shift. In this chapter, an atomistic spring-mass based finite element approach appropriate to simulate the vibration characteristics of single-walled and multi-walled CNTs is presented. The method uses spring-mass finite elements that simulate the interatomic interactions and the inertia effects in CNTs. It uses a special technique for simulating the bending between adjacent bonds, distinguishing it from other mechanics-based methods. This method utilizes the exact microstructure of the CNTs while allowing the straightforward input of the force constants that molecular theory provides. In addition, spring-like elements are formulated in order to simulate the interlayer van der Waals interactions. These elements connect all atoms between different CNT layers at a distance smaller than the limit below which the van der Waals potential tends to zero. The related stiffness is a function of this distance. The resulting dynamic equilibrium equations can be used to generate new results. Results available in the literature were used to validate the proposed method. Parametric analyses are performed reporting the natural frequencies as well as the mode shapes of various multi-walled CNTs for different aspect geometric characteristics. Furthermore, the principle of mass detection using resonators is based on the fact that the resonant frequency is sensitive to the resonator mass, which includes the self-mass of the resonator and the attached mass. The change of the attached mass on the resonator causes a shift to the resonant frequency. Since, the key issue of mass detection is in quantifying the change in the resonant frequency due to the added mass, the effect of added mass to the vibration signature of CNTs is investigated for the clamped-free and clamped-clamped support conditions. And different design parameters. Additionally, the frequency shifts of single- and multi-walled CNTs were compared. 2. CNTs geometry A planar layer of carbon atoms forms a periodic structure called the graphene sheet. Pencil lead consists of a stack of overlaying graphene sheets that easily separate upon shearing in Advances in Vibration Analysis Research 118 writing. A perfect graphene sheet in the xy-plane consists of a doubly periodic hexagonal lattice defined by two base vectors, ( ) 1,0a= 1 v and () 13 cos60 ,sin60 , 22 oo aa ⎛⎞ == ⎜⎟ ⎜⎟ ⎝⎠ 2 v (1) where α is equal to 3 h r and h r is the radius of the hexagonal cell. Note that the lengths of these vectors are equal. Any point of plane (,)Pxy = is uniquely defined as a linear combination of these two vectors, 1 ab=+ + 02 Pv v v, (2) where a and b are integers, provided that 0 v is the center of a hexagon. Knowing the geometry of graphene, a single-walled CNT can be geometrically generated by rolling a single-layer graphene sheet, which is ideally cut, to make a cylinder. The graphene sheet must be rolled up in the direction of the chiral vector h C defined as (see Figure 1): 12 nm = + h Caa (3) where 1 a and 2 a are the basis vectors of the honeycomb lattice and integers ( n , m ) are the number of steps along the zigzag carbon bonds and generally are used to name a nanotube. Fig. 1. Generation of a SWCNT from a graphene sheet. [...]... for vibration analysis First, in order to validate the proposed method, we compare the results obtained from the present method with outputs from other theoretical approaches based on molecular or 126 Advances in Vibration Analysis Research continuum mechanics, as shown in Table 1 The comparison is limited in terms of different sequence of vibration modes This is mainly because of differences in L... maximum increase in the radius is observed on the half length of the tube An Atomistic-based Spring-mass Finite Element Approach for Vibration Analysis of Carbon Nanotube Mass Detectors 127 (a) (b) (c) Fig 6 Radial-like modes of vibration of clamped-clamped supported single-walled CNTs: (a) radial breathing mode, (b) triangular mode, and (c) cross mode 128 Advances in Vibration Analysis Research. .. mechanical properties of single walled carbon nanotubes using a spring based finite element approach, Computational Materials Science 41 (2008) 56 1 -56 9 [18] S.K Georgantzinos, G.I Giannopoulos, N.K Anifantis, An efficient numerical model for vibration analysis of single-walled carbon nanotubes, Computational Mechanics 43 (2009) 731-741 [19] S.K Georgantzinos, N.K Anifantis, Vibration analysis of multi-walled... 80 1208 793 7880.9 7693 6 58 57 9 56 30 5 C-F 10 8 52 51 8 50 21 17 1 21 228 3907 26907 1.76 2 20 260 6940 30420 20 190 2800 19390 20 170 2320 16700 1.09 0.41 2.44 3 4 C-C Middle Table 2 Comparison of the frequency shift between single- and multi-walled CNTs with the same lengths and diameters 4 .5 Comparison between single and multi-walled CNT mass detectors The main issue to examine is whether multi-walled... 362 (2004) 20 65- 2098 [4] K Jensen, K Kim, D.A Zettl, An atomic-resolution nanomechanical mass sensor, Nature Nanotechnology 3 (2008) 53 3 -53 7 [5] R.G Knobel, Weighing single atoms with a nanotube, Nature Nanotechnology 3 (2008) 52 5 -52 6 [6] R Mateiu, A Kuhle, R Marie, A Boisen, Building a multi-walled carbon nanotube-based mass sensor with the atomic force microscope, Ultramicroscopy 1 05 (20 05) 233–237... Spring-mass Finite Element Approach for Vibration Analysis of Carbon Nanotube Mass Detectors 137 [10] B Liu, Y Huand, H Jiang, S Qu, K.C Hwang, The atomic-scale finite element method, Computer methods in applied mechanics and engineering, 193 (2004) 1849-1864 [11] B Liu, Y Huand, H Jiang, S Qu, K.C Hwang, The atomic-scale finite element method, Computer methods in applied mechanics and engineering,... This is mainly because of differences in L (nm) Support Condition of inner CNT Support Condition of outer CNT 1.1 4.1 C-C C-C 0.9478 0.9276 [13] 0.4 1.1 5. 5 C-C C-C 0.6410 0.7 355 [13] 0.4 1.1 8.0 C-C C-C 0. 355 1 0.3323 [13] 0.7 1.4 14 Free C-C 0. 158 2 0.16 65 [ 25] 0.7 1.4 14 Free C-F 0.0288 0.0270 [ 25] 0.7 1.4 14 C-C C-C 0.1661 0.1718 [ 25] 0.7 1.4 20 C-C C-C 0.04 ~0.03 [24] Di (nm) Do (nm) 0.4 Fundamental... described in [17], where γ = 30 o in the hexagonal lattice of the graphene This angle may vary for each C-C-C microstructure in a CNT according to its type and radius due to its cylindrical shape In the case of chiral nanotubes, the stiffness of the three different bending springs (Figure 4(b)) varies kb 1 ≠ kb 2 ≠ kb 3 In the cases of armchair and zigzag nanotubes, two of the three bending spring stiffnesses... accompanied by a simultaneous movement of all atoms in the longitudinal direction and changing tube length during the vibration 4.3 Effect of layers on CNT vibration In order to investigate the influence of the number of layers on the vibration characteristics of a nanotube, CNTs of the same aspect ratio (i.e., the same length and outer diameter) were chosen for analysis with the proposed technique Figure... 107 106 1 05 0 1 2 Frequency (THz) (a) 1012 m / mr = 102 x / L = 1.0 x / L = 0 .5 1011 Amplitude (nm) 1010 109 108 107 106 1 05 0 1 2 Frequency (THz) (b) Fig 11 Vibration spectra of clamped-free supported (5, 0) CNT with a mass m/mr = 100 attached on the (a) x/L = 0. 75 and (b) x/L = 0 5 position compared with CNT tip case An Atomistic-based Spring-mass Finite Element Approach for Vibration Analysis of . 1.1 5. 5 C-C C-C 0.6410 0.7 355 [13] 0.4 1.1 8.0 C-C C-C 0. 355 1 0.3323 [13] 0.7 1.4 14 Free C-C 0. 158 2 0.16 65 [ 25] 0.7 1.4 14 Free C-F 0.0288 0.0270 [ 25] 0.7 1.4 14 C-C C-C 0.1661 0.1718 [ 25] . to a laminated plate with ceramic core as a host plate. Advances in Vibration Analysis Research 114 8. Conclusion In this paper free vibration of a thin FGM plus piezoelectric laminated. consists of a stack of overlaying graphene sheets that easily separate upon shearing in Advances in Vibration Analysis Research 118 writing. A perfect graphene sheet in the xy-plane consists