Nonlinear Dynamic Response of Carbon Nanotube Nanocomposite Microbeams

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Marek Cetraro ◽  
Walter Lacarbonara ◽  
Giovanni Formica
Keyword(s):  
Carbon Nanotubes ◽  
Dynamic Response ◽  
Nonlinear Dynamic ◽  
Volume Fraction ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Path Following ◽  
Transversely Isotropic ◽  
Nonlinear Frequency ◽  
Nonlinear Dynamic Response

The nonlinear dynamic response of nanocomposite microcantilevers is investigated. The microbeams are made of a polymeric hosting matrix (e.g., epoxy, polyether ether ketone (PEEK), and polycarbonate) reinforced by longitudinally aligned carbon nanotubes (CNTs). The 3D transversely isotropic elastic constitutive equations for the nanocomposite material are based on the equivalent inclusion theory of Eshelby and the Mori–Tanaka homogenization approach. The beam-generalized stress resultants, obtained in accordance with the Saint-Venant principle, are expressed in terms of the generalized strains making use of the equivalent constitutive laws. These equations depend on both the hosting matrix and CNTs elastic properties as well as on the CNTs volume fraction, geometry, and orientation. The description of the geometry of deformation and the balance equations for the microbeams are based on the geometrically exact Euler–Bernoulli beam theory specialized to incorporate the additional inextensibility constraint due to the relevant boundary conditions of microcantilevers. The obtained equations of motion are discretized via the Galerkin method retaining an arbitrary number of eigenfunctions. A path following algorithm is then employed to obtain the nonlinear frequency response for different excitation levels and for increasing volume fractions of carbon nanotubes. The fold lines delimiting the multistability regions of the frequency responses are also discussed. The volume fraction is shown to play a key role in shifting the linear frequencies of the beam flexural modes to higher values. The CNT volume fraction further shifts the fold lines to higher excitation amplitudes, while it does not affect the backbones of the modes (i.e., oscillation frequency–amplitude curves).

Geometrical Nonlinear Effects in the Dynamic Response of Sandwich Beams

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Itay Odessa ◽  
Oded Rabinovitch ◽  
Yeoshua Frostig
Keyword(s):  
Dynamic Response ◽  
Nonlinear Dynamic ◽  
High Order ◽  
Theory Approach ◽  
Nonlinear Frequency ◽  
Sandwich Beams ◽  
Nonlinear Dynamic Response ◽  
The Core ◽  
Geometrical Nonlinear ◽  
The Face

Abstract The geometrical nonlinear dynamic response of sandwich beams is studied using a dynamic high-order nonlinear model. The model is derived using the variational principle of virtual work and uses the Extended High-Order Sandwich Panel Theory approach with consideration of two interfaces between the three layers. A first-order shear deformation theory is adopted for the face sheets, while the kinematic assumption of high-order small deformations that account for out-of-plane compressibility are considered for the core layer. The nonlinearity of the dynamic model is introduced by considering geometrically nonlinear kinematic relations in the face sheets. The nonlinear kinematic relations and the dynamic modeling aim to evaluate the effects of the two features and their coupling on the response. The nonlinear dynamic response of sandwich beams is studied through two numerical cases and comparison of the nonlinear results with their linear counterparts. The first case looks into the coupling of the global geometrical nonlinear behavior with the dynamic behavior. The second case focuses on the local instability of the face sheets and its interaction with the compressibility of the core in the dynamic response of soft core sandwich beams. The comparison of linear and nonlinear dynamic response in the two cases sheds light on the coupling of the geometrical nonlinear and dynamic behaviors. Among other features, the latter is expressed by nonlinear attractors, higher modes response, nonlinear frequency response, and significant wrinkling response.

The Selection of the Linearized Natural Frequency for the Second-Order Normal Form Method

2011 ◽  
Author(s):  
Zhenfang Xin ◽  
S. A. Neild ◽  
D. J. Wagg
Keyword(s):  
Normal Form ◽  
Dynamic Response ◽  
Natural Frequency ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Second Order ◽  
Nonlinear Dynamic Response ◽  
Weakly Nonlinear ◽  
Selection Of ◽  
Normal Form Technique

The normal form technique is an established method for analysing weakly nonlinear vibrating systems. It involves applying a simplifying nonlinear transform to the first-order representation of the equations of motion. In this paper we consider the normal form technique applied to a forced nonlinear system with the equations of motion expressed in second-order form. Specifically we consider the selection of the linearised natural frequencies on the accuracy of the normal form prediction of sub- and superharmonic responses. Using the second-order formulation offers specific advantages in terms of modeling lightly damped nonlinear dynamic response. In the second-order version of the normal form, one of the approximations made during the process is to assume that the linear natural frequency for each mode may be replaced with the response frequencies. Here we will show that this step, far from reducing the accuracy of the technique, does not affect the accuracy of the predicted response at the forcing frequency and actually improves the predictions of sub and superharmonic responses. To gain insight into why this is the case, we consider the Duffing oscillator. The results show that the second-order approach gives an intuitive model of the nonlinear dynamic response which can be applied to engineering applications with weakly nonlinear characteristics.

Nonlinear Dynamic Response of Piezoelectric Cylindrical Shell with Delamination under Hygrothermal Conditions

2012 ◽  
Vol 204-208 ◽  
pp. 4698-4701
Author(s):  
Jin Hua Yang ◽  
De Liang Chen
Keyword(s):  
Cylindrical Shell ◽  
Finite Difference Method ◽  
Dynamic Response ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Transverse Motion ◽  
Governing Equations ◽  
Nonlinear Dynamic Response ◽  
The Finite Difference Method ◽  
Delamination Length

Abstract. On the basis of the nonlinear plate-shell and piezoelectric theory, the governing equations of motion for axisymmetrical piezoelectric delaminated cylindrical shell under hygrothermal conditions were derived. The governing equation of transverse motion was modified by contact force and thus the penetration between two delaminated layers could be avoided. The whole problem was resolved by using the finite difference method. In calculation examples, the effects of delamination length, depth and amplitude of load on the nonlinear dynamic response of the axisymmetrical piezoelectric delaminated shell under hygrothermal conditions were discussed in detail.

scholarly journals Nonlinear Dynamic Response of Functionally Graded Rectangular Plates under Different Internal Resonances

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Y. X. Hao ◽  
W. Zhang ◽  
X. L. Ji
Keyword(s):  
Dynamic Response ◽  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Internal Resonances ◽  
Third Order ◽  
Nonlinear Dynamic Response

The nonlinear dynamic response of functionally graded rectangular plates under combined transverse and in-plane excitations is investigated under the conditions of 1 : 1, 1 : 2 and 1 : 3 internal resonance. The material properties are assumed to be temperature-dependent and vary along the thickness direction. The thermal effect due to one-dimensional temperature gradient is included in the analysis. The governing equations of motion for FGM rectangular plates are derived by using Reddy's third-order plate theory and Hamilton's principle. Galerkin's approach is utilized to reduce the governing differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms, which are then solved numerically by using 4th-order Runge-Kutta algorithm. The effects of in-plane excitations on the internal resonance relationship and nonlinear dynamic response of FGM plates are studied.

Semi-exact solution for nonlinear dynamic analysis of nanobeams reinforced with functionally graded carbon nanotube located on a viscoelastic foundation

2019 ◽  
Vol 233 (18) ◽  
pp. 6626-6639
Author(s):  
Ahmad Fallah ◽  
MB Dehkordi ◽  
YT Beni
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Dynamic Response ◽  
Dynamic Analysis ◽  
Nonlinear Dynamic ◽  
Nonlinear Coefficient ◽  
Nonlinear Dynamic Analysis ◽  
Functionally Graded ◽  
Nonlinear Frequency ◽  
Viscoelastic Foundation

In this investigation, a transient nonlinear dynamic analysis of nanobeams reinforced with carbon nanotubes, which is located on a nonlinear viscoelastic foundation under the impulse loading, is investigated. The boundary conditions of the nanobeam are considered as clamped-clamped, and the carbon nanotube is distributed in different distribution along the thickness of nanobeam. First, using the Hamiltonian method and taking advantage of the couple stress theory and considering the Von Karman relationship between strain and displacement, the differential equation governing for Euler–Bernoulli nanobeam is obtained. Then, by using the semi-exact method and the Galerkin's method, the displacement derivatives are separated from the time derivatives and the equation derived is solved using Runge–Kutta's numerical method. In order to confirm the equation and its solution, a comparative study is performed that shows an appropriate fitting between the results. Finally, the influence of parameters such as nonlinear coefficient of foundation, applied force, size effect, and type of nanotube distribution on the nonlinear frequency to linear frequency ratio and transient nanobeam dynamic response is investigated. A study is also conducted on the effect of foundation damping coefficient and the inclusion of nonlinear effects on the transient dynamic response when the nanobeam is under impulse load and resonance conditions. The results show that the nonlinear vibrational frequency of the nanobeam with the FG-X carbon nanotube distribution is the highest, and the FG-O carbon nanotubes distribution is the least.

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