## 2 Statement of the Modal Modification Problem
### Governing Equation  
In the case of a thermoelastically coupled system, the eigenproblem becomes complex and is the main concern here. For problems considering thermoelastic damping, the thermal dilation term is taken into account in the equation of momentum. The equations of motion for the generalized thermodynamic problem can be formulated in matrix form as follows:  
\[\mathbf{M}\mathbf{\ddot{u}}+\mathbf{K}\mathbf{u}-\mathbf{G}\mathbf{\Theta}= \mathbf{0} \tag{1}\]  
where \(\mathbf{u}\) is the displacement vector and \(\mathbf{\Theta}\) is the temperature vector. In this study, we consider that \(\mathbf{M}\) is a positive definite lumped mass matrix and \(\mathbf{K}\) is a non-negative definite symmetric stiffness matrix. The lumped mass matrix is a diagonal positive definite matrix obtained from the traditional consistent mass matrix [36; 37]. The lumped mass matrix is widely used in finite element dynamical analyses, and its accuracy has been validated in numerical examples [38; 39]. The diagonal elements of the lumped matrix are rational to  
Figure 1: Sketch map of the regime of thermoelastic damping (TED). As the clamped-clamped microbeam resonator vibrates in a purely flexural mode, one side along the thickness of the beam stretches, which causes the temperature of the region to decrease. The temperature of the compressed side increases, causing a temperature gradient along the thickness of the beam. The irreversible heat flux induced by the temperature gradient causes the dissipation of energy. the corresponding diagonal elements of the original consistent mass matrix. Furthermore, the sums of all elements of the two mass matrices are assumed to be equal such that  
\[\mathbf{M}_{ii}^{\text{lumped}}=\frac{\mathbf{M}_{ii}^{\text{consistent}}}{ \sum\mathbf{M}_{ii}^{\text{consistent}}}\sum\sum\mathbf{M}_{ij}^{\text{consistent}} \tag{2}\]  
where \(\sum\) is a conventional operator for the summation of the elements in the matrix. We will use \(\mathbf{M}\) for the lumped matrix hereafter for simplicity. The matrix of thermal dilation \(\mathbf{G}\) is the coupled term in the equation of motion. Note that even if the lumped mass matrix is singular (e.g., it can be encountered in finite element analysis involving shell elements), the mass matrix can also be formed to be non-singular by introducing the Guyan reduced method [40]. On the other hand, the governing equation of heat conduction takes the following form:  
\[\mathbf{S}\mathbf{\Theta}+\mathbf{H}\mathbf{\dot{\Theta}}+\mathbf{F}\mathbf{ \dot{u}}=\mathbf{0} \tag{3}\]  
where \(\mathbf{S}\), \(\mathbf{H}\) and \(\mathbf{F}\) are the matrices of thermal conductivity, heat generation, and heat generation coupled by deformation, respectively. Similar to the mass matrix in the governing equation of motion, the heat generation matrix \(\mathbf{H}\) in the equation of heat conduction is constructed to be a positive definite lumped matrix. A more detailed derivation of the governing equations of the thermodynamically coupled system can be found in [32] and [7]. Notice that both the equations of motion and heat conduction contain the coupled terms. This indicates that they have to be solved simultaneously for the fully coupled system. To overcome the effect of higher-order time derivatives in the system equations of motion and heat conduction, a transformation is applied to the system equations to reduce the order of the thermoelastically coupled system, despite the increased dimensionality of the eigenvalue problem [41]. The traditional direct combination of governing equations is listed below [7]:  
\[ \begin{bmatrix}\mathbf{H}&\mathbf{F}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{M}\\ \mathbf{0}&\mathbf{I}&\mathbf{0}\end{bmatrix}\begin{bmatrix}\mathbf{\dot{ \Theta}}\\ \mathbf{\dot{u}}\\ \mathbf{\ddot{u}}\end{bmatrix} + \begin{bmatrix}\mathbf{S}&\mathbf{0}&\mathbf{0 }\\ -\mathbf{G}&\mathbf{K}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&-\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{\Theta} \\ \mathbf{u}\\ \mathbf{\dot{u}}\end{bmatrix}