A Unified Analysis for the Free Vibration of the Sandwich Piezoelectric Laminated Beam with General Boundary Conditions under the Thermal Environment

Table 1

Boundary conditions and stiffness matrixes.

Boundary conditions	Essential conditions	Stiffness matrixes
F	Nxx = Qxz = Mxx = 0	diag (0, 0, 0, 0)
S	u0 =  = Mxx = 0	diag (10^18, 10^18, 0, 0)
C	u0 =  = θ = 0	diag (10^18, 10^18, 10^18, 0)
E	u0, , θ ≠ 0	diag (ku, , kθ, kΦ)
E1	u0 = θ = 0,	diag (10^18, , 10^18, 0)
E2	u0 =  = 0, θ ≠ 0	diag (10^18, 10^18, kθ, 0)
E3	u0 = 0, , θ ≠ 0	diag (10^18, kw, kθ, 0)