Nonlinear vibration analysis of nanotubes has fascinated strong attention from the researchers due to their stiffest and strongest known fibers with excellent tensile strength. However, the purpose of using nanotubes as a partial replacement for reinforcement material has become a novelty in tomorrow’s applications. Nanotube reinforcement composites (NTRC), which are created by initiating aligned nanotubes to the polymer matrix will likely expand the application of nanotubes. Some researched investigations have been outlined to divulge mechanical behaviors of (NTRC) structures. modeled the nonlinear partial differential equation of motion by presenting a vibration analysis of SWCNTs using the Hamiliton principle, Euler-Bernoulli beam theory, and Eringen's non-local elastic theory. The model is then resolved using the cosine after treatment approach and the differential transformation method. studied free vibrational analysis of embedded SWCNTs under longitudinal magnetic field based on non-local strain gradient theory and Euler-Bernoulli beam model. They also used Galerkin and the equivalent linearization methods to obtained stability. Thereafter, simulation was conducted on the parameters in the nonlinear model. investigated the free vibrations of SWCNTs under the theory of non-local elasticity, and the differential quadrature method is used to solve the resultant nonlinear partial differential equation. analyzed the dynamic response of a single-walled carbon nanotube under Euler-Bernoulli beam theory and Eringen's non-local elastic theory with various boundary conditions. After applying dimensionless parameters to the duffing equation of motion, the governing non-linear mathematical model is broken down using the Galerkin decomposition approach. The Homotropy Perturbation Method is then used to solve the resultant nonlinear duffing equation. investigated SWCNTs' nonlinear force vibration under the influence of the Casimir force using Hamiltonian principles and Eringen non-local theory. After applying the Galerkin decomposition approach to the resultant nonlinear duffing equation, a nonlinear mathematical model was created and solved using the Homotropy Perturbation approach. Since they argue for both the concrete's compression and tensile zones, the stability and dynamic response simulation findings turn out to be accurate. examined the stability and flow-induced vibrations of SWCNTs placed in biological soft tissue using the Kelvin-Voigt foundation. The differential transformation technique is used to solve the nonlinear problem that is created using Hamilton's concept. created a non-local strain gradient anisotropic elastic shell nonlinear model to address the vibration of pinned-pinned supports SWCNTs under the Mindlin strain gradient theory, Eringen nonlocal elasticity theory, and Sanders-Koiter shell theory for strain-displacement. Consequently, the parameters are used in the numerical simulations. examined stepped nano-beams immersed in the elastic medium using nonlinear free vibration analysis based on Hamilton's principle and Eringen's nonlocal elasticity theory. Using the perturbation approach once more, the mathematical model was solved. The outcome showed that the elastic medium coefficient decreased vibrational amplitudes, while the nonlocal parameter had a negative influence on frequency. determined how boundary circumstances affected the SWCNT's free vibrations. Following that, the nonlinear governing equation is modeled using Flügge's shell dynamic equations. Additionally, the vibrational frequency equation is determined in typical eigenvalue form using the wave propagation technique. The findings are displayed visually and contrasted with those found in the literature. employed the differential transform method and the cosine after-treatment technique to analyze the nonlinear vibrations of a SWNT with multiple layers on elastic media in a thermal-magnetic environment using the Euler-Bernoulli beam model and nonlocal elasticity theory. This method proved to be very useful for designing CNTs in thermal and magnetic environments. provided a study of laminated composite panels that are flat, concave, and convex and reinforced with carbon nanotubes. Consequently, they analyzed the model using the numerical homogenization of finite element technique, and the outcomes are contrasted with those of traditional composite materials according to various end circumstances. examined the vibrational behavior of fluidic conveying SWCNTs using the differential transform technique and the theories of thermo-elastic mechanics, non-local elasticity, and Euler-Bernoulli beams to describe the system that rested on an elastic medium with clamped-clamped and pinned-pinned supports. discussed using the continuum technique for free vibrational analysis of pinned-pinned supports, including SWCNTs and DWCNTs. Therefore, in the free vibrational analysis of single-walled and double-walled carbon nanotubes, the accuracy of beam models and standard 2D shell models. investigated the nonlinear vibration analysis of a single-walled carbon nanotube with a longitudinal magnetic field utilizing Eringen's Non-local Elasticity theory and Euler-Bernoulli Beam theory to create a nonlinear model, which is then solved using the finite element technique. examined single-wall carbon nanotubes with metallic, semi-conducting, and (6, 5) chirality molar absorbance coefficients. In addition to being independent of the diameter distribution and the electrical kinds of single wall carbon nanotubes, the majority of coefficients have values that fall between 2 and 5 × 109 /mL mol^-1 cm^-1. Lastly, their coefficient is independent of chirality and electrical kinds and is reliant on the length of single wall carbon nanotubes. examined free vibrational analysis of Single-Walled Carbon Nanotubes with exponentially changing stiffness under nonlocal and Euler-Bernoulli beam theory to model the system, and the differential quadrature method was used to solve the model after converting it to a generalized eigenvalue problem. The results demonstrate that, in contrast to other literature, the solutions obtained using the differential quadrature method converge quickly. examined the various stability of SWCNTs in chiral, armchair, and zigzag configurations lying on elastic media exposed to both high and low temperatures. Once more, surface factors like surface energy and surface residual stresses are taken into account while modeling, and the outcomes are confirmed by another research. examined SWCNT vibrational analysis under a single variable shear deformation beam theory, while also subjected to thermal conditions and an axial or longitudinal magnetic field. The gradient, non-local, and Hamilton's principle are used to build the mathematical model. examined the vibrations of implanted multi-walled branching carbon nanotubes with nanofluid conveyance on Winkler-Pasternak medium in a thermal-magnetic environment. The nonlinear governing mathematical model is developed using Euler-Bernoulli beam theory, Hamilton's principle, and nonlocal elasticity theory. It is then tied to the energy equations of the vibrational models and Navier-Stoke's equations using multi-dimensional numerical PDE solvers in conjunction with MATLAB PDE tools. The parameters in the solved duffing equation are also subjected to numerical simulations in accordance with this.

Figure 1 illustrates the problem formulation, which is a single-walled carbon nanotube supported by a linear and nonlinear Winklers and Pasternak foundation. The problem is modeled using Hamilton's principle, Euler-Bernoulli beam theory, and Eringen's nonlocal elasticity beam theory. The following is the longitudinal magnetic field:

The Winklers foundation, both linear and nonlinear, indicates, However, = , where is linear Winkler-type and is non-linear Winkler-type. Substituting back into equation (1) gives:

The concept also incorporates a Pasternak-type foundation that serves as; , initial stresses as , parameter induced by residual surface stress as and change in temperature is as well as the electrostatic force acting on the system, van der Waal force and Casimir force which are denoted as; .and . Substituting back into equation (2) yield;

Where, Electrostatic force, van der Waal force and Casimir force are given as:

Substituting them into equation (3) becomes;

When equation (4)'s right-hand side is solved, the result is:

Through expanding series solution, equation (5)'s right hand side (RHS) becomes:

Substituting equation (6), (7), (8), (9) and (10) into equation (5) produce;

The parameters in the model are defined as:

𝐼: Second moment of inertia

𝐸: Young modulus of elasticity

𝑚: Tube mass per unit length

♒: Magnetic field permeability

: Magnetic field strength

𝐴: Cross sectional area

𝑘: Foundation stiffness

: Nonlocal material constant

𝑎: Internal characteristic length of the nanotube

𝑤: Displacement

𝐿: Length of the tube

𝑡: Time

𝑥: Cartesian axis

: Plank constant

c: Speed of sound

b: Width of the tube

d: Diameter of the tube

Nevertheless, the following displacement boundary conditions are applied to this investigation of the dynamic response of single-walled carbon nanotubes:

Pinned-Pinned supported (P-P) nanotube,

Clamped-Clamped supported (C-C) nanotube,

Clamped-Pinned supported (C-P) nanotube,

Clamped-Free supported (C-F) nanotube,

Using a few dimensionless parameters, equation (11) is conveniently rearranged into dimensionless form. Included in these dimensionless variables are:

Equation (17) below is the dimensionless equation of motion when Eq. (16) is substituted into equation (11), yield:

The Galerkin decomposition approach is used to transform Eq. (17) into a nonlinear duffing equation. As demonstrated, this process enables the deflection of the SWCNT’s to be expressed as a product of two independent functions.

Where a function , chosen to meet the end supports conditions requirements is shown in Table 1. Remember that the Galerkin transform, one parameter at a time, is defined as follows:

The nonlinear equation is . The decomposition equation is as follows when Eq. (19) is applied to the governing equations:

After integrating and combining the temporal component as a coefficient, the SWCNT's Duffing equation is as follows:

Expanding equation (21), gives;

Subject to ,

Equation (22) coefficients are:

Iteration and perturbation techniques are coupled to create the Iteration Perturbation Method. This approach works for differential equations that are substantially nonlinear as well as mildly nonlinear issues. The nonlinear equation (22), for example, requires the introduction of a fake parameter ε in order to provide an iteration perturbation solution of the governing equation.

Using the iterative perturbation method on equation (22)'s a duffing equation and dividing the result by m yields:

However, let replace some parameters in equation (31) as:

Substituting equation (32) into equation (31) gives;

The governing equation's iteration perturbation solution can be obtained by introducing an artificial parameter ε in the following manner:

Equation (34) can be approximated by:

Where , is the first approximation answer. With ω representing the unknown angular frequency, the first solution can be considered to be as follows: . Substituting into equation (36) yields:

It can be further simplified as:

The equation (38) can be written in the following form after application of combine trigonometry:

assuming that,

Substituting equation (40) and (41) into equation (39) after picking in power of one yield:

If the term exists in equation (42), the final solution will contain the secular term, . As a result, this term's coefficient in (42) ought to be zero, giving us:

Solving the above equation (43), yields:

From equation (41), setting

Substituting into equation (45), gives:

Nonetheless the fundamental frequency of vibration is obtained as:

Consequently, the frequency-ratio is obtained as;

Substituting value of gives;

Solving equation (42) with secular terms and boundary conditions, is obtained as:

Adding the obtained series solutions together equation (51) yield;

And also substituting gives:

Therefore, deflection of four boundary conditions considered are obtained as stated below:

When simple-simple supports, (C-C) are used, deflection turns into;

When clamped-clamped supports, (C-C) are used, deflection turns into;

When clamped-simple supports, (C-C) are used, deflection turns into;

When clamped-free supports, (C-F) are used, deflection turns into;

The scientific analysis of the simulated outcome using the answers found in chapter three is the main topic of this part.

Figures 2, 3, 4, and 5 illustrate how Casimir force is affected by the dynamic response analysis of single-wall carbon nanotubes in a temperature, magnetic, and electrostatic environment. Indicating the structure's propensity to deform at particular natural frequencies, they exhibit deflections at five different buckling and mode forms.

Figures 6, 7, 8, and 9 illustrate how the thermal term of various end conditions affects the dimensionless amplitude-frequency ratio curve of the dynamic response analysis of single-wall carbon nanotubes with surface effect and geometric imperfection that are supported by elastic foundations in a thermal-magnetic-electrostatic environment while being affected by Casimir force. On the other hand, Figure 9 shows that the rise from zero to maximum frequency response reduces as well, even if the thermal term grows from zero to maximum frequency response above linear system. These demonstrate that in order to stabilize the structure of SWCNTs, the thermal term in Figures 6, 7, and 8 must be maintained at their minimums, while the thermal term in Figure 9 must be maintained at its maximum.

Figures 10, 11, 12, and 13 illustrate how various magnetic term boundary conditions affect the dimensionless amplitude-frequency ratio curve of the dynamic response analysis of single-wall carbon nanotubes with surface effect and geometric imperfection that are supported by elastic foundations in a magneto-thermal-electrostatic environment while Casimir force is present. The figures show that the magnetic term raises the frequency response above the linear system from zero to maximum, while Figure 13 shows that the frequency response lowers as well. These demonstrate that in order to stabilize the structure of SWCNTs, the magnetic term in Figures 10, 11, and 12 must be maintained at both minimums, whereas the magnetic term in Figure 13 must be maintained at its maximum.

Figures 14, 15, 16, and 17 illustrate how various boundary conditions of the linear Winkler elastic foundation affect the dimensionless amplitude-frequency ratio curve of the dynamic response analysis of single-wall carbon nanotubes with surface effect and geometric imperfection resting on elastic foundations in a magneto-thermal-electrostatic environment while being affected by Casimir force. Nevertheless, the data show that the frequency-ratio falls toward a linear system as the linear Winkler elastic medium grows from zero to maximum, indicating that the structure must be maintained at maximum in order to acquire stability. Furthermore, because clamped-free foundations are becoming more linear, they are a better choice for this kind of building.

Figures 18, 19, 20, and 21 illustrate how the Pasternak elastic medium with various boundary conditions affects the dynamic response analysis's dimensionless amplitude-frequency ratio curve for single-wall carbon nanotubes with surface effect and geometric imperfection that are supported by elastic foundations in a magneto-thermal-electrostatic environment while being affected by Casimir force. As seen in Figures 18, 19, and 20, the Pasternak elastic medium grows from zero to maximum frequency response, surpassing the linear system. However, Figure 21 shows a drop in frequency response as well. These demonstrate that in order to stabilize the structure of SWCNTs, the Pasternak foundation for Figures 18, 19, and 20 must be maintained at both minimum and maximum, but Figure 21's foundation must be maintained at maximum.

The impact of various boundary conditions of the nonlinear Winkler elastic medium on the dimensionless amplitude-frequency ratio curve of vibration of single-walled carbon nanotubes resting on an elastic foundation with thermal and magnetic effects under the influence of electrostatic force is depicted in Figures 22, 23, 24, and 25. Figures 22, 23, 24, and 25 show that the frequency-ratio grows in proportion to the nonlinear Winkler-type elastic medium's growth from zero to maximum. This demonstrates that a minimal nonlinear Winkler elastic basis is required for the structure of SWCNTs to become stable.

Under the influence of electrostatic force, Figures 26, 27, 28, and 29 illustrate how the nonlocal parameter of various boundary conditions affects the dimensionless amplitude-frequency ratio curve of vibration of single-walled carbon nanotubes resting on an elastic foundation with magnetic and thermal effects. when seen in Figures 26, 27, 28, and 29, the frequency-ratio also falls when the nonlocal parameter rises from zero to maximum. Nevertheless, in order to ensure stability for all of the end circumstances taken into account during simulations, the nonlocal value should be kept as low as possible.

Under the influence of Casimir force, Figures 30, 31, 32, and 33 illustrate how the electrostatic force of various boundary conditions affects the dimensionless amplitude-frequency ratio curve of the dynamic response analysis of single-wall carbon nanotubes with surface effect and geometric imperfection resting on elastic foundations in a magneto-thermal-electrostatic environment. According to the numbers, the frequency-ratio increases as electrostatic force grows from zero to maximum. However, in order to establish stability, the electrostatic force should be reduced to a minimum.

Van der Waals force of various boundary conditions is shown to have an impact on the dimensionless amplitude-frequency ratio curve of dynamic response analysis of single-wall carbon nanotubes with surface effect and geometric imperfection resting on elastic foundations in a magneto-thermal-electrostatic environment under the influence of Casimir force in Figures 34, 35, 36, and 37. According to the numbers, the frequency-ratio rises as the van der Waal force grows from zero to maximum. Nonetheless, to maintain stability, the van der Waals force should be minimized.

In the dynamic response analysis of single-wall carbon nanotubes with surface effect and geometric imperfection resting on elastic foundations in a magneto-thermal-electrostatic environment, the effect of different boundary conditions' Casimir forces on the dimensionless amplitude-frequency ratio curve is seen in Figures 38, 39, 40, and 41. The data show that the frequency-ratio grows as the Casimir force rises from zero to maximum. To maintain stability, the Casimir force should be minimized in this way.

The dynamic response analysis of single-wall carbon nanotubes with geometric imperfections resting on elastic foundations in a magneto-thermal-electrostatic environment under the influence of Casimir force is shown in Figure 42, where the impact of four boundary conditions is depicted. The maximum frequency ratio is shown by clamped-free, clamped-pinned, and pinned-pinned supports. The lowest frequency ratio, however, is seen in clamped-clamped support. Because of this, it is possible to manage the stability of any elastic foundation when a system is subjected to Casimir force by using clamped-clamped support, which exhibits the best foundation types with the lowest frequency ratio.

The dynamic response analysis of single-wall carbon nanotubes with geometric imperfections resting on elastic foundations in a magneto-thermal-electrostatic environment under the influence of Casimir force is shown in Figure 44, where the impact of two boundary conditions is depicted. The maximum frequency ratio is shown by clamped-free, clamped-pinned, and pinned-pinned supports in Figure 44. The lowest frequency ratio, however, is seen in clamped-clamped support. Because of this, it is possible to manage the stability of any elastic foundation when a system is subjected to Casimir force by using clamped-clamped support, which exhibits the best foundation types with the lowest frequency ratio.

The dynamic response analysis of single-wall carbon nanotubes with geometric imperfections resting on elastic foundations in a magneto-thermal-electrostatic environment under the influence of Casimir force is shown in Figure 43, where the impact of three boundary conditions is depicted. The maximum frequency ratio is shown by clamped-free, clamped-pinned, and pinned-pinned supports in Figure 43. The lowest frequency ratio, however, is seen in clamped-clamped support. Because of this, it is possible to manage the stability of any elastic foundation when a system is subjected to Casimir force by using clamped-clamped support, which exhibits the best foundation types with the lowest frequency ratio.

The dynamic response analysis of single-wall carbon nanotubes with geometric imperfections resting on elastic foundations in a magneto-thermal-electrostatic environment under the influence of Casimir force is shown in Figure 45, where the impact of one boundary conditions is depicted. The maximum frequency ratio is shown by clamped-clamped. The stability of any elastic foundation can be managed when a system is subjected to Casimir force by using clamped-clamped support, which exhibits the best foundation type.

A dynamic response study of single-wall carbon nanotubes with geometric imperfections lying on elastic foundations in a magneto-thermal-electrostatic environment under the effect of Casimir force is shown in Figures 46, 47, 48, and 49. The time-deflection curve's deflection across each end supports the corresponding decrease towards the crest as the Casimir force grows from zero to maximum. Therefore, when the forces grow from zero to maximum in the case of clamped-free supports, the deflection trough changes into a deflection crest after the first two simulations.

Van der Waal force has an impact on the time-deflection curve of the dynamic response analysis of single-wall carbon nanotubes with geometric imperfections resting on elastic foundations in a magneto-thermal-electrostatic environment under the influence of Casimir force, as shown in Figures 50, 51, 52, and 53. As van der Waal force increases from zero to maximum, the deflection trough of the time-deflection curve for each end support considered decreases towards the crest, so that in the case of clamped-free supports, following the first two simulations, the deflection trough changes into the deflection crest.

Electrostatic force's effect on the time-deflection curve of the dynamic response analysis of single-wall carbon nanotubes with geometric imperfections resting on elastic foundations in a magneto-thermal-electrostatic environment under the influence of Casimir force is depicted in Figures 54, 55, 56, and 57. The deflection trough of the time-deflection curve for either end supports the corresponding drop towards the crest as electrostatic force grows from zero to maximum. As the forces grow from zero to maximum, the deflection trough in the case of clamped-free supports changes to a deflection crest after the first two simulations.

In a magneto-thermal-electrostatic environment with Casimir force acting on single-wall carbon nanotubes with geometric imperfections resting on elastic foundations, Figures 59, 59, 60, and 61 illustrate how the mass of the nanotube affects the time-deflection curve of the dynamic response analysis. As mass of nanotube increases from 0.10kg to maximum, deflection crest of time-deflection curve for each ends supports considered decrease respectively towards trough. Again, when the mass of nanotube increases from minimum to maximum, deflection crest reduces as well, indicating that as the mass increases, deflection decreases for all ends conditions studied.

Using Eringen's non-local elastic theory, Euler-Bernoulli elastic beam theory, and the Hamiliton principle as a basis, this work examines the nonlinear vibration analysis of single-wall carbon nanotubes with geometric imperfections resting on elastic foundations in a magneto-thermal-electrostatic environment under the influence of Casimir Force. The nonlinear mathematical model is solve using Iteration Perturbation Method after Galerkin decomposition method is applied directly to decomposed the system into spatial and temporal nonlinear duffing equation of motion. Again, MATLAB simulation was conducted on the analyzed solution as the results obtained shows influence of Casimir force, electrostatic force, and van der Waals force and mass of the nanotubes as synonymous as there contribute significantly to enhanced both compressive and tensile zones of reinforced concrete.

The authors declare no conflicts of interest.