Nonlinear Vibrations of Multiwalled Carbon Nanotubes under Various Boundary Conditions

Table 1

Common boundary conditions for the transverse vibration of beams.

End conditions of beam	Mode shape (normal function)	Value of 𝛽
	sin𝛽𝑥𝐿	𝜋
	cosh𝛽𝑥𝐿−cos𝛽𝑥𝐿	4.730041
	−cosh𝛽−cos𝛽sinh𝛽−sin𝛽sinh𝛽𝑥𝐿−sin𝛽𝑥𝐿
	cosh𝛽𝑥𝐿−cos𝛽𝑥𝐿	3.926602
	−cosh𝛽−cos𝛽sinh𝛽−sin𝛽sinh𝛽𝑥𝐿−sin𝛽𝑥𝐿