# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 4 Equations of Motion  
The equations of motion for the finite element model are obtained by means of the [PERSON]'s principle [41]:  
\[\int_{0}^{T}\left[\delta K-\left(\delta U+\delta V\right)\right]\,dt=0, \tag{23}\]  
where \(\delta K\) represents the virtual kinetic energy, \(\delta U\) is the virtual strain energy, and \(\delta V\) is the virtual potential energy due to the external loads.  
[PERSON]'s principle allows us to directly derive the finite element equations (i.e., the mass matrix is obtained with the virtual kinetic energy, the stiffness matrix with the virtual strain energy, and the load vector with the virtual potential energy). For natural vibration, it is only necessary to determine the stiffness and mass matrices. The matrix calculation procedure is explained in Section 5.  
The virtual kinetic energy is given by  
\[\delta K=\int_{V}\,\rho\delta\dot{\mathbf{u}}\cdot\dot{\mathbf{u}}\,dV, \tag{24}\]  
where \(\rho\) is the density. Substituting the displacement field and the differential volume element \(dV=JdAd\varsigma^{3}\) in the above equation, expressions for the mass matrix coefficients can be obtained. Integrating along \(\varsigma^{3}\) yields  
\[\delta K = \int_{A}\,\left\{I_{0}\,(\delta\dot{\mathbf{u}}\cdot\dot{\mathbf{ u}})\,+\,I_{1}\left[\frac{h}{2}\,(\dot{\mathbf{u}}\cdot\delta\dot{\mathbf{\varphi}}+ \dot{\mathbf{\varphi}}\cdot\delta\dot{\mathbf{u}})\right]+I_{2}\left[\frac{h}{2}\, \left(\dot{\mathbf{u}}\cdot\delta\dot{\mathbf{\varphi}}\mathbf{n}+\frac{h}{2}\, \dot{\mathbf{\varphi}}\cdot\delta\dot{\mathbf{\varphi}}+\dot{\Psi}\mathbf{n}\cdot \delta\dot{\mathbf{u}}\right)\right]+\right. \tag{25}\] \[\left.+\,I_{3}\left[\frac{h^{2}}{4}\,\left(\dot{\mathbf{\varphi}} \cdot\delta\dot{\Psi}\mathbf{n}+\dot{\Psi}\mathbf{n}\cdot\delta\dot{\mathbf{ \varphi}}\right)\right]+I_{4}\left[\frac{h^{2}}{4}\left(\dot{\Psi}\delta\dot{ \Psi}||\mathbf{n}||^{2}\right)\right]\right\}dA,\]  
where \(I_{0},I_{1},I_{2},I_{3},I_{4}\) are the mass inertias defined as follows:  
\[I_{0}=\int_{-1}^{1}\rho\,Jd\varsigma^{3}, I_{1}=\int_{-1}^{1}\rho\,\varsigma^{3}\,Jd\varsigma^{3}, I_{2}=\int_{-1}^{1}\rho\,\left(\varsigma^{3}\right)^{2}\,Jd \varsigma^{3},\] \[I_{3}=\int_{-1}^{1}\rho\,\left(\varsigma^{3}\right)^{3}\,Jd \varsigma^{3}, I_{4}=\int_{-1}^{1}\rho\,\left(\varsigma^{3}\right)^{4}\,Jd \varsigma^{3},\]  
and \(||\mathbf{n}||\) is the norm of the normal vector \(\mathbf{n}\).  
The virtual strain energy is computed by  
\[\delta U=\int_{V}\delta\mathbf{E}\cdot\mathbf{E}:\mathbf{C}\,dV. \tag{26}\]  
Substituting the Green-Lagrange tensor and the fourth-order elasticity tensor, an expression to determine the stiffness matrix is obtained:  
\[\delta U=\int_{A}\left[A^{ijkl}\delta\varepsilon_{ij}^{(0)}\varepsilon_{kl}^{ (0)}+B^{ijkl}\,\left(\delta\varepsilon_{ij}^{(1)}\varepsilon_{kl}^{(0)}+ \delta\varepsilon_{ij}^{(0)}\varepsilon_{kl}^{(1)}\right)+D^{ijkl}\delta \varepsilon_{ij}^{(1)}\varepsilon_{kl}^{(1)}\right]\,Jd\varsigma^{3}dA, \tag{27}\]  
where the contravariant components \(A^{ijkl},B^{ijkl},D^{ijkl}\) are the effective extensional, extensional-bending coupling, and bending fourth-order stiffness tensor components [14]. These components are calculated as  
\[A^{ijkl}=\int_{-1}^{1}C^{ijkl}\,Jd\varsigma^{3}, B^{ijkl}=\int_{-1}^{1}\varsigma^{3}C^{ijkl}\,Jd\varsigma^{3}, D^{ijkl}=\int_{-1}^{1}\left(\varsigma^{3}\right)^{2}C^{ijkl}\,Jd \varsigma^{3}.\]