. Author manuscript; available in PMC: 2019 Mar 1.

Published in final edited form as: J Mech Phys Solids. 2017 Dec 8;112:187–208. doi: 10.1016/j.jmps.2017.12.002

Table 1.

The analytical model for the first-order out-of-plane vibration of the buckled ribbon

Vibration mode

Normalized linear natural
frequency

Simplified form for A₍₀₎ ≪ h

U_{1} = \frac{1}{2} [1 + cos (\frac{2 π Z}{L})] U_{3} \frac{π A_{(0)}}{2 L} sin (\frac{4 π Z}{L})

{\hat{f}}_{I} = \frac{\sqrt{6} π}{3} \frac{A_{(0)}}{h}

Simplified form for A₍₀₎ ≫ h

U_{1} = \frac{1}{2} [1 - cos (\frac{4 π Z}{L})] U_{3} = \frac{π A_{(0)}}{3 L} [6 sin (\frac{2 π Z}{L}) - 2 sin (\frac{6 π Z}{L})]

{\hat{f}}_{I} = 2 \sqrt{3} π \sqrt{\frac{3 + 8 π^{2} (A_{(0)}^{2} / L^{2})}{27 + 40 π^{2} (A_{(0)}^{2} / L^{2})}} = 2 \sqrt{3} π \sqrt{\frac{3 + 8 ε_{compre}}{27 + 40 ε_{compre}}}

Precise form

U_{1} = C (\frac{Δ A_{(2)}}{Δ A_{(1)}}) {[1 + cos (\frac{2 π Z}{L})] + \frac{Δ A_{(2)}}{Δ A_{(1)}} [1 - cos (\frac{4 π Z}{L})]} U_{3} = C (\frac{Δ A_{(2)}}{Δ A_{(1)}}) {\frac{π A_{(0)}}{2 L} sin (\frac{4 π Z}{L}) + \frac{π A_{(0)}}{3 L} \frac{Δ A_{(2)}}{Δ A_{(1)}} [6 sin (\frac{2 π Z}{L}) - 2 sin (\frac{6 π Z}{L})]}

{\hat{f}}_{I} (\frac{A_{(0)}}{h}, \frac{A_{(0)}}{L})

Here, C(ΔA₍₂₎/ΔA₍₁₎) is a single-variable function to normalize the mode, as presented in Appendix C.