## 3 Statement of the Problem  
The illustration in Figure 2 displays a very thin, flexible microbeam with a constant cross-section \(A\) that has been excited by an ultrashort laser. The \(x\) directions specify the beam axis, while the \(y\) and \(z\) variables specify the microbeam width and thickness, respectively. The microbeam is exposed to small-amplitude bending oscillations along the \(x\)-axis, resulting in a deflection that conforms to the linear Euler-Bernoulli theory (EBT). As shown below, the EBT with small deflections \(w(x,t)\) is used to figure out the displacements and stresses. \[u=-z\left(\frac{\partial w}{\partial x}\right),v=0,w=w(x,t) \tag{7}\]  
\[e_{xx}=e=-z\frac{\partial^{2}w}{\partial x^{2}} \tag{8}\]  
The Pasternak type that has two parameters is the greatest straightforward extension of the Winkler model what has a single parameter. It functions by attaching the ends of the springs to a plate that employs a shear layer, which is thought to be incompressible and simulates the shear interaction that occurs between the various spring components. The microbeam that is under consideration is contained within a homogeneous, three-parameter, viscoelastic substrate. The Pasternak (shear) basis modulus \(K_{p}\), the damping  
Figure 2: Thin, flexible beam resting on visco-Pasternak’s foundation that is induced by an ultrafast laser pulse. modulus \(K_{c}\) of the viscous medium and, finally, the linear Winkler modulus \(K_{s}\) create the foundation model. Taking into account the fact that there is infinite contact between the microbeam and the material, the reaction behaves according to a basic three-parameter visco-Pasternak model, according to the relationship. \[R_{f}(x,t)=K_{S}w-K_{p}\frac{\partial^{2}w}{\partial x^{2}}+K_{C}\frac{ \partial w}{\partial t} \tag{9}\]  
This model may be reduced to the visco-Winkler model by setting \(K_{p}\) equal to zero. It is possible to eliminate the viscoelastic term by setting \(K_{c}=0\). The is one constitutive equation for a one-dimensional thermoelastic medium, which is provided by:  
\[\sigma_{x}=E\frac{\partial u}{\partial x}-\alpha_{T}E\theta=-E\bigg{[}z\frac {\partial^{2}w}{\partial x^{2}}+\alpha_{T}\theta\bigg{]} \tag{10}\]  
where \(\alpha_{T}=\alpha_{t}/(1-2\
u)\), \(\alpha_{t}\) denotes thermal expansion, E represents the modulus of elasticity, \(\
u\) denotes the Poisson's ratio, and \(\sigma_{x}\) indicates the thermal stress. The equation of motion for the microscale beams can be stated as follows when using [PERSON]'s basis and [PERSON]'s principle and taking into account the presence of an axial force:  
\[\frac{\partial^{2}M}{\partial x^{2}}=\rho A\frac{\partial^{2}w}{\partial t^{2 }}+\sigma_{0}A\frac{\partial^{2}w}{\partial x^{2}}+R_{f}(x,t) \tag{11}\]  
where \(M\) represents the moment of bending, which may be calculated using the following equation:  
\[M=\int z\sigma_{x}dA \tag{12}\]  
When Equation (10) is put into Equation (12), the bending moment may be calculated as  
\[M(x,t)=-EI\bigg{[}\frac{\partial^{2}w}{\partial x^{2}}+\alpha_{T}M_{T}\bigg{]} \tag{13}\]  
in which the cross-section inertia moment is \(I=bh^{3}/12\), and \(M_{T}=\frac{12}{h^{3}}\int_{-h/2}^{h/2}\theta(x,z,t)\mathrm{d}z\) stands for the thermal moment.