Table 2.

Values of c₁, c₂, c₃, c₄, ζ_i, and λ_i for different boundary conditions.

Boundary Condition	c 1	c 2	c 3	c 4	ζi	λi
C–C	1	−1	1	−1	$\frac{\cosh \left({\lambda }_i\right)-\cos \left({\lambda }_i\right)}{\sinh \left({\lambda }_i\right)-\sin \left({\lambda }_i\right)}$	$\cosh \left({\lambda }_i\right)\cdot \cos \left({\lambda }_i\right)=1$
C–SS	1	−1	1	−1	$\frac{\cosh \left({\lambda }_i\right)-\cos \left({\lambda }_i\right)}{\sinh \left({\lambda }_i\right)-\sin \left({\lambda }_i\right)}$	$\tan \left({\lambda }_i\right)=\tanh \left({\lambda }_i\right)$
F–F	1	1	1	1	$\frac{\cosh \left({\lambda }_i\right)-\cos \left({\lambda }_i\right)}{\sinh \left({\lambda }_i\right)-\sin \left({\lambda }_i\right)}$	$\cosh \left({\lambda }_i\right)\cdot \cos \left({\lambda }_i\right)=1$
C–F	1	−1	1	−1	$\frac{\sinh \left({\lambda }_i\right)-\sin \left({\lambda }_i\right)}{\cosh \left({\lambda }_i\right)+\cos \left({\lambda }_i\right)}$	$\cosh \left({\lambda }_i\right)\cdot \cos \left({\lambda }_i\right)=-1$
F–SS	1	1	1	1	$\frac{\cosh \left({\lambda }_i\right)-\cos \left({\lambda }_i\right)}{\sinh \left({\lambda }_i\right)-\sin \left({\lambda }_i\right)}$	$\tan \left({\lambda }_i\right)=\tanh \left({\lambda }_i\right)$