The Size-Dependent Thermoelastic Vibrations of Nanobeams Subjected to Harmonic Excitation and Rectified Sine Wave Heating

In this article, a nonlocal thermoelastic model that illustrates the vibrations of nanobeams is introduced. Based on the nonlocal elasticity theory proposed by Eringen and generalized thermoelasticity, the equations that govern the nonlocal nanobeams are derived. The structure of the nanobeam is under a harmonic external force and temperature change in the form of rectified sine wave heating. The nonlocal model includes the nonlocal parameter (length-scale) that can have the effect of the small-scale. Utilizing the technique of Laplace transform, the analytical expressions for the studied fields are reached. The effects of angular frequency and nonlocal parameters, as well as the external excitation on the response of the nanobeam are carefully examined. It is found that length-scale and external force have significant effects on the variation of the distributions of the physical variables. Some of the obtained numerical results are compared with the known literature, in which they are well proven. It is hoped that the obtained results will be valuable in micro/nano electro-mechanical systems, especially in the manufacture and design of actuators and electro-elastic sensors.


Introduction
In recent decades, due to the rapid advancement in engineering technology and stringent training requirements, dynamics and stability and their control over mechanical vibration have gradually transformed into a fundamental and indivisible branch of study in applied mechanics and related engineering. Mechanical vibration is considered in many conditions and circumstances as a useful phenomenon, used in many areas and can also better serve people's lives. Vibration analysis is fundamental for developments, as well as structural and mechanical system design. These data support us to predict the performance of the structure under various external loads and to design a control system, which is used to analyze the development vibrations in a cantilever beam instead of the structure itself [1].
The field of micro-electro-mechanical systems (MEMSs) is fast becoming involved in many resistance and correspondence applications. Modern technologies have been created to manufacture a variety of MEMS gadgets to meet the demand for many precision industries. MEMSs consist of elastic mechanical parts, such as micro-bridges, cantilevers, and microscopic films of various geometrical sizes and engineering forms that often contain loads [2]. It is imperative for MEMS designers to understand the mechanical properties of elastic micro-components, considering the ultimate goal of equation depends on the thermal relaxation times [25] that are applied. The Laplace transform procedure is utilized as part of the deduction. The effects due to the harmonic external load, nonlocal and angular frequency parameters are represented graphically and are investigated. The current model can be used in micro/nano-electro-mechanical applications, such as mass flow sensors, accelerometers, relay switches, frequency filters and resonators. The vibration of nanobeams is a significant topic for the study of nanotechnology, as it relates to the optical and electronic properties of the nanobeams.

Theoretical Problem Formulations
The schematic representation of the considered system is illustrated in Figure 1, showing a hinged-hinged nanobeam of length L (0 ≤ x ≤ L), width b (−b/2 ≤ y ≤ b/2) and thickness h (−h/2 ≤ z ≤ h/2) and Young's modulus E, assuming that the Euler-Bernoulli beam theory is employed for modeling the nanobeams and the cross section of the nanobeam is uniform along the entire length. Hence, the displacements of the beam can be written as where w(x, t) denotes the transverse displacement (lateral deflection) of the nanobeam. behavior of axial nanomaterial systems, such as nanoplates or nanoscale rods with thermoelastic properties. In this investigation, the thermoelastic nonlocal theory is applied to the Euler Bernoulli beam problem subjected to a dispersed harmonic excitation load per unit length. The non-Fourier conduction equation depends on the thermal relaxation times [25] that are applied. The Laplace transform procedure is utilized as part of the deduction. The effects due to the harmonic external load, nonlocal and angular frequency parameters are represented graphically and are investigated. The current model can be used in micro/nano-electro-mechanical applications, such as mass flow sensors, accelerometers, relay switches, frequency filters and resonators. The vibration of nanobeams is a significant topic for the study of nanotechnology, as it relates to the optical and electronic properties of the nanobeams.

Theoretical Problem Formulations
The schematic representation of the considered system is illustrated in Figure 1, showing a hinged-hinged nanobeam of length L (0 ≤ x ≤ L), width b (−b/2 ≤ y ≤ b/2) and thickness h (−h/2 ≤ z ≤ h/2) and Young's modulus E, assuming that the Euler-Bernoulli beam theory is employed for modeling the nanobeams and the cross section of the nanobeam is uniform along the entire length. Hence, the displacements of the beam can be written as where w(x, t) denotes the transverse displacement (lateral deflection) of the nanobeam. Based on Eringen's nonlocal elasticity theory [14,15], the constitutive relation for a one-dimensional problem, after using Equation (11), can be written as where σ x is the nonlocal axial stress, θ = T − T 0 is the resonator temperature change, T is the distribution of temperature and T 0 denotes the environmental temperature, α T = α t /(1 − 2ν), ν is Poisson's ratio and α t is the linear thermal expansion. In Equation (2), the nonlocal parameter is ξ = (e 0 ) 2 , where is the internal characteristic length and e 0 is a suitable parameter that can be determined by the experiment. When the characteristic parameter is neglected, then ξ = 0 , and Equation (2) reduces to the classical constitutive relation (local elasticity).
The bending moment M(x, t) is given by Based on Eringen's nonlocal elasticity theory [14,15], the constitutive relation for a one-dimensional problem, after using Equation (11), can be written as

Substitution of Equation (2) into Equation (3) results in
where σ x is the nonlocal axial stress, θ = T − T 0 is the resonator temperature change, T is the distribution of temperature and T 0 denotes the environmental temperature, α T = α t /(1 − 2ν), ν is Poisson's ratio and α t is the linear thermal expansion. In Equation (2), the nonlocal parameter is ξ = (e 0 a) 2 , where a is the internal characteristic length and e 0 is a suitable parameter that can be determined by the experiment. When the characteristic parameter a is neglected, then ξ = 0, and Equation (2) reduces to the classical constitutive relation (local elasticity).
The bending moment M(x, t) is given by where I = bh 3 /12 and M T is the thermal moment defined by

Substitution of Equation (2) into Equation (3) results in
When the beam is due to a distributed load q(x, t), the transverse motion equation is as follows [24]: where ρ is the density of the material and A is the cross section of the nanobeam. Substitution of Equation (4) into Equation (6) results in Substituting Equation (7) into Equation (6), Equation (6) can be rewritten in the form The non-Fourier heat transfer equation, considering the entropy balance proposed by Lord and Shulman [25], which includes the heat flux in addition to its time derivative, is given as where K is the coefficient of thermal conductivity, Q is the heat source, C E denotes the specific heat at constant strain, τ 0 is the relaxation time characteristic according to the Lord and Shulman theory and e = ∂u ∂x is the volumetric strain. Substituting Equation (1) into the heat equation, Equation (9), when Q = 0, is given by

Solution of the Problem
We assume that the increment of temperature change is in terms of a sine function (sinusoidal variation) as Using relation (11) in the governing Equations (7), (8) and (10), we obtain Introducing the following non-dimensional quantities: x , w , u , z , L , h , b = ηc{x, w, u, z, L, h, b}, t , τ q , τ θ = ηc 2 t, τ q , τ θ , Substitution of (15) into (12)- (14) results in (dropping the primes for convenience) In general, the harmonic excitation q(x, t) has the form of a sine or cosine function of a single frequency. In this problem, the dynamic load q(x, t) can be considered as where F 0 is the magnitude of forcing excitation and Ω is the frequency of the external excitations (Ω = 0 for the uniformly distributed load).

Boundary Conditions
Let us also consider what the boundary conditions are assumed to be, as in Table 1   Table 1. The boundary conditions of the problem.
x Mechanical Boundary Conditions Thermal Boundary Conditions In Table 1, the function f(x, t) is a varying rectified sine wave function which is described mathematically as where Θ 0 is a constant and ω is the frequency of the rectified sine wave.

Solution in the Transformed Space
In this problem, the initial conditions are assumed to be Employing the technique of Laplace transform to Equations (16)- (18), we obtain where  (22) and (23), we can obtain where The general solution to Equation (26) can be presented as: where L j and M j are the integral parameters and ±m j , (j = 1, 2, 3) satisfy the equation From Equations (26) and (25), we can obtain The general solutions of Equation (10) with the help of Equation (30) can be simplified as where Introducing the solutions of w and Θ into Equation (24) yields The axial displacement u is obtained using Equations (28) and (1) as Using the Laplace transform domain, the boundary conditions given in Table 1 may be written as It is difficult to obtain direct Laplace transform inversion for the complicated solutions of the transformed studied fields. Thus, the physical solutions are obtained numerically using the Fourier expansion technique [26]. In this method, any function g(x, s) can be transformed to that in the domain of time g(x, t) by the relation: where N f is a finite number and the parameter c satisfies the relation ct 4.7 [27].
In the calculation analysis, we take L/h = 10 and b/h = 0.5 and the parameter ξ (ξ = 10 2 ξ) is also considered. The effects of several effective parameters on the conduct of the deflection of the nanobeam are explored. The numerical results are plotted in Figures 2-4, which explain the variations of the distributions of lateral vibration, displacement, temperature and moment, concerning different parameters. α T = 2.59 × 10 (1/K), ν = 0.22, K = 156 W/(mK).
In the calculation analysis, we take L/h = 10 and b/h = 0.5 and the parameter ξ ̅ (ξ ̅ = 10 2 ξ) is also considered. The effects of several effective parameters on the conduct of the deflection of the nanobeam are explored. The numerical results are plotted in Figures 2-4, which explain the variations of the distributions of lateral vibration, displacement, temperature and moment, concerning different parameters.   Figure 2a, when ξ is increased from 0 to 3.0, the dynamic response of the deflection w changes from a softening to a stiffness type behavior. It is also detected that the deflection vibration of the nanobeam corresponds to the midpoint and meets the boundary condition at the two ends x = 0, L. Figure 2b demonstrates the temperature profile of the thermoelastic nanobeam for different values of the nanoscale parameter ξ. From the figure, it is found that that temperature θ decreased over time, which means the mechanical energy of the nanobeam is wasted in the form of thermal energy. Additionally, by increasing the values of ξ, we note a decrease in the temperature profile.
From Figure 2c, it is observed that an increase in the value of parameter ξ results in an increase in the dynamic displacement u profile. The bending moment M increases as the distance x increases, as is shown in Figure 2d. It can be seen from Figure 2d that the bending moment M increases with an increase in the nonlocal parameter ξ. In previous research [23,24], they found that all the studied fields of the nanobeam clearly depend on the nanoscale parameter ξ. Now, our calculation demonstrates that this phenomenon is also valid for various values of the parameter ξ. In Figure 3a-d the variation of the studied fields of the beam with respect to the magnitude of the forcing excitation F 0 is shown. In the following case, we take the values of the parameters ξ, ω and Ω, respectively, as {1, 3, 5} [26]. One can see from these figures that the effect of the forcing excitation F 0 has pronounced influences on the distribution of all studied fields. Additionally, it can be seen that the increase in the value of the forcing excitation F 0 causes an increase in the thermodynamic temperature values, displacement and the lateral fields, which is evident in the peak points of the profiles. This result is consistent with the results of [18,20]. It is clear from Figure 3d    The variation in the dimensionless field quantities with the external excitation frequency Ω are shown in Figure 4a-d. The value of Ω = 0 indicates the static forcing excitation, while other values indicate the harmonic forcing excitation. It is apparent from these figures that, when the excitation frequency coefficient is increased, the amplitude of the studied fields increases. The data would seem to suggest that increasing the amount of angular frequency causes a decrease in both the static equilibrium position and the amplitude of the response. When an external harmonic excitation is applied to the nanobeam, the field quantities are more sensitive to the excitation frequency Ω. The results obtained in our research are found to be in good agreement with [18]. In Harrington et al.'s research [29], they found that the temperature changes with the resonance frequency of the external excitation.  Figure 2a, when ξ ̅ is increased from 0 to 3.0, the dynamic response of the deflection w changes from a softening to a stiffness type behavior. It is also detected that the deflection vibration of the nanobeam corresponds to the midpoint and meets the boundary condition at the two ends x = 0, L . Figure 2b demonstrates the temperature profile of the thermoelastic nanobeam for different values of the nanoscale parameter ξ ̅ . From the figure, it is found that that temperature θ decreased over time, which means the mechanical energy of the nanobeam is wasted in the form of thermal energy. Additionally, by increasing the values of ξ ̅ , we note a decrease in the temperature profile.
From Figure 2c, it is observed that an increase in the value of parameter ξ ̅ results in an increase in the dynamic displacement u profile. The bending moment M increases as the distance x increases, as is shown in Figure 2d. It can be seen from Figure 2d that the bending moment M increases with an increase in the nonlocal parameter ξ ̅ . In previous research [23,24], they found that all the studied fields of the nanobeam clearly depend on the nanoscale parameter ξ ̅ . Now, our calculation demonstrates that this phenomenon is also valid for various values of the parameter ξ ̅ .
In Figure 3a-d the variation of the studied fields of the beam with respect to the magnitude of the forcing excitation F 0 is shown. In the following case, we take the values of the parameters ξ ̅ , ω The influence of the angular frequency of the rectified sine wave ω of the varying rectified sine wave heating on the vibrations of the studied fields of the nanobeam under harmonic external excitation is presented in Figure 5a-d. The small scale and other parameters ξ, F 0 and Ω are assumed to be constant in this case. The numerical results of the studied fields are displayed for various angular frequency values ω = 1, 2, 3. It is observed that the parameters of the angular frequency ω affect the characteristics of the nanobeam significantly. From Figure 5a-d, it is found that the amplitude of temperature, deflection and axial displacement, as well as the bending moment, decrease with increasing time. excitation is presented in Figure 5a-d. The small scale and other parameters ξ ̅ , F 0 and Ω are assumed to be constant in this case. The numerical results of the studied fields are displayed for various angular frequency values ω = 1,2,3. It is observed that the parameters of the angular frequency ω affect the characteristics of the nanobeam significantly. From Figure 5a-d, it is found that the amplitude of temperature, deflection and axial displacement, as well as the bending moment, decrease with increasing time.

Conclusions
In the current investigation, the thermoelastic vibration of nanobeams under the effect of a harmonic external force and rectified sine wave heating is discussed. Using the nonlocal elasticity theory and non-Fourier heat conduction model, basic equations are derived. The effect of different parameters, such as the nonlocal parameter ξ, the magnitude of forcing excitation F 0 , the angular frequency of thermal vibration ω and the frequency of the external excitation Ω on the studied fields of the nanobeam was additionally examined. From the obtained results, we found that: - The magnitude of forcing excitation, the nonlocal parameter, the frequency of the external excitation and the frequency of the rectified sine wave heating field have a considerable influence on the response of the system behavior. -When an external harmonic excitation is applied to the nanobeam, the field quantities are more sensitive to excitation frequencies. - The increase in the value of the forcing excitation causes an increase in the thermodynamic temperature values, displacement and in the lateral fields, which is evident in the peak points of the profiles. - The results in this study may find applications and requests in the development and design of resonators due to thermal environmental loading.
-The vibration of the heat response of nanobeams can vary by changing the size of the external force and thermal loads without having to change any other engineering and physical parameters of the nanobeams. - The temperature dependence of the frequency of the resonance of resonator devices can be utilized to design and create precision thermometers that are not at all suitable under high sensitivity to rectified sine wave heating and harmonic loads.