Abstract
Purpose
This paper reported the modeling steps of coupled beam nanocomposite system and its fundamental frequency characteristics. The model has been derived for two straight coupled beams (made of carbon nanotube, CNT) as the continuous transitional springs, including a middle elastic layer.
Methods
The effective properties of the beam components are evaluated based on the well-known rule of mixture. Additionally, to achieve the generality of the material distribution, various configurations of CNT have been adopted in the current model through the thickness of the structure. The main objective of the present study is to evaluate the effect of material temperature dependency on the frequency parameter of the nanocomposite double beam system. The structural model has been derived mathematically using the robust lower-order theory (first-order shear deformation, FSDT) to compute the eigenvalues without hampering the degree of solution accuracies. The nanotube properties are also considered to be temperature-dependent (TD). The final eigenvalues are computed via a semi-analytical approach, namely generalized differential quadrature (GDQ) approach.
Results
First, the natural frequencies of homogenous beam system are compared with the results of frequency parameter available in the literature. Then, the fundamental frequencies of the coupled beam system have been computed analytically for different structural input parameters (material properties, end constraints of full structure and middle elastic layer) to showcase the effect individual and/or combined conditions with and without the influence of temperature.
Conclusion
The final outcomes are revealed in detail for different input parameters and the subsequent outcomes.
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Data Availability
The data that support the findings of this study are available from the corresponding author, upon reasonable request.
Abbreviations
- \(V_{{{\text{CN}}}}^{{}}\) :
-
Volume of CNTs in beam
- \(V_{{{\text{M}}^{^{\prime}} }}\) :
-
Volume of matrix in beam
- \(E_{{{\text{CN}}}}^{11} \;{\text{and}}\;E_{{{\text{CN}}}}^{22}\) :
-
CNTs Young’s modulus
- \(G_{{{\text{CN}}}}^{12}\) :
-
CNTs shear modulus
- \(\upsilon_{{{\text{CN}}}}\) :
-
CNTs Poisson ratio
- \(\rho_{{{\text{CN}}}}\) :
-
CNTs density
- \(E_{{\text{M}}}^{{}}\) :
-
Matrix Young’s modulus
- \(G_{{\text{M}}}^{{}}\) :
-
Matrix shear modulus
- \(\upsilon_{{\text{M}}}\) :
-
Matrix Poisson ratio
- \(\rho_{M}\) :
-
Matrix density
- \(E_{{\text{R}}}^{11} \;{\text{and}}\;E_{{\text{R}}}^{22}\) :
-
Equivalent Young’s modulus
- \(G_{{\text{R}}}^{12}\) :
-
Equivalent shear modulus
- \(\upsilon\) :
-
Equivalent Poisson ratio
- \(\rho\) :
-
Equivalent density
- \(\eta_{1} ,\,\eta_{2} ,\,\eta_{3}\) :
-
Size-dependent parameters
- \(D_{1}^{i} \;{\text{and}}\;D_{2}^{i}\) :
-
Field displacement
- \(d_{1}^{i} \;{\text{and}}\;d_{2}^{i}\) :
-
Neutral axis displacement
- \(\sigma_{x}^{i}\) :
-
Normal stress
- \(\tau_{xz}^{i}\) :
-
Shear stress
- \(E\) :
-
Elastic modulus
- \(\nu\) :
-
Poisson’s ratio
- \(N_{x}^{i}\) :
-
Normal force
- \(Q_{x}^{i}\) :
-
Shear force
- \(M_{x}^{i}\) :
-
Bending moment
- \(\omega\) :
-
Natural frequency of system
- \(\overline{\omega }\) :
-
Dimensionless natural frequency of system
- \(D_{{{\text{eff}}}}\) :
-
Effective stiffness matrix
- \(M_{{{\text{eff}}}}\) :
-
Mass matrix
- \(D_{{\text{I}}}\) :
-
Stiffness matrix related to internal points
- \(D_{{{\text{IB}}}}\) :
-
Internal stiffness matrix associated with BCs
- \(D_{{\text{B}}}\) :
-
BCs matrix stiffness
- \(\left\{ {X_{{\text{I}}} } \right\}\) :
-
Vector related to internal grid points
- \(\left\{ {X_{{\text{B}}} } \right\}\) :
-
Vector related to BCs
- K :
-
Stiffness of mid-layer springs
- \(k_{{\text{s}}}\) :
-
Stiffness factor
- T :
-
Environment temperature
- \(T_{0}\) :
-
Room temperature (= 300° k)
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Ghandehari, M.A., Masoodi, A.R. & Panda, S.K. Thermal Frequency Analysis of Double CNT-Reinforced Polymeric Straight Beam. J. Vib. Eng. Technol. 12, 649–665 (2024). https://doi.org/10.1007/s42417-023-00865-0
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DOI: https://doi.org/10.1007/s42417-023-00865-0