# Correction to: Crack mathematical modeling to study the vibration analysis of cracked micro beams based on the MCST

Correction

## 1 Correction to: Microsystem Technologies  https://doi.org/10.1007/s00542-018-3768-7

In the original publication of this article, the Eqs. (20), (21), (52), (54), (55), (56) and Figs. 4–15 were incorrectly published. The author would like to correct them as follows:

The Eqs. (20), (21), (52), (54), (55), and (56) should be corrected as follows:
$$U_{c} = \frac{{\left( {1 - \vartheta^{2} } \right)bh}}{E}\mathop \int \limits_{0}^{\eta } (K_{IM} )^{2} d\eta$$
(20)
$$C = \left[ {1 + \frac{12}{{\left( {1 + \vartheta } \right)\left( {1 - \eta } \right)^{2} }} \left( {\frac{l}{h}} \right)^{2} } \right]\left[ {\frac{{\left( {1 - \vartheta^{2} } \right)bh}}{E} \frac{{\partial^{2} }}{{\partial M^{2} }}\mathop \int \limits_{0}^{\eta } (K_{IM} )^{2} d\eta } \right]$$
(21)
$$\frac{{dw_{2} }}{dx}\left( {L_{c} } \right) - \frac{{dw_{1} }}{dx}\left( {L_{c} } \right) = \frac{{d^{2} w_{1} }}{{dx^{2} }}\left( {L_{c} } \right) \times \frac{S}{{K_{t} }}$$
(52)
$$Q_{61} = \beta \cos \left( {\beta L_{c} } \right) - \frac{{S\beta^{2} }}{{K_{t} }}\sin \left( {\beta L_{c} } \right) ; \quad Q_{62} = - \beta \sin \left( {\beta L_{c} } \right) - \frac{{S\beta^{2} }}{{K_{t} }}\cos \left( {\beta L_{c} } \right)$$
$$Q_{63} = \beta \cosh \left( {\beta L_{c} } \right) + \frac{{S\beta^{2} }}{{K_{t} }}\sinh \left( {\beta L_{c} } \right) ; Q_{64} = \beta \sinh \left( {\beta L_{c} } \right) + \frac{{S\beta^{2} }}{{K_{t} }}\cosh \left( {\beta L_{c} } \right)$$
(54)
$$C = \frac{{\left( {1 - \vartheta^{2} } \right)bh}}{E} \frac{{\partial^{2} }}{{\partial M^{2} }}\mathop \int \limits_{0}^{\eta } (K_{IM} )^{2} d\eta$$
(55)
$$K_{t} = \frac{1}{C} = \left[ {\frac{{\left( {1 - \vartheta^{2} } \right)bh}}{E} \frac{{\partial^{2} }}{{\partial M^{2} }}\mathop \int \limits_{0}^{\eta } (K_{IM} )^{2} d\eta } \right]^{ - 1}$$
(56)
Also, Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15 should be corrected as follows: