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

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:

Fig. 4

Torsional spring stiffness at the crack location of the microbeam with considering the SIF in model No. 1 for crack versus crack depth ratio $\eta = a/h$ for different values of dimensionless material length scale parameter $l/h$

Fig. 5

Torsional spring stiffness at the crack location of the microbeam with considering the SIF in model No. 2 for crack versus crack depth ratio $\eta = a/h$ for different values of dimensionless material length scale parameter $l/h$

Fig. 6

Torsional spring stiffness at the crack location of the microbeam with considering the SIF in model No. 3 for crack versus crack depth ratio $\eta = a/h$ for different values of dimensionless material length scale parameter $l/h$

Fig. 7

Torsional spring stiffness at the crack location of the microbeam with considering the SIF in model No. 4 for crack versus crack depth ratio $\eta = a/h$ for different values of dimensionless material length scale parameter $l/h$

Fig. 8

Torsional spring stiffness at the crack location of the microbeam versus crack depth ratio $\eta = a/h$ for different models of crack at $\frac{l}{h} = 0$

Fig. 9

Torsional spring stiffness at the crack location of the microbeam versus crack depth ratio $\eta = a/h$ for different models of crack at $\frac{l}{h} = 0.5$

Fig. 10

Torsional spring stiffness at the crack location of the microbeam versus crack depth ratio $\eta = a/h$ for different models of crack at $\frac{l}{h} = 1$

Fig. 11

Variation of fundamental natural frequency of the cracked microbeam versus crack depth ratio $\eta = a/h$ for different crack location at $\frac{l}{h} = 1$

Fig. 12

Variation of second natural frequency of the cracked microbeam versus crack depth ratio $\eta = a/h$ for different crack location $\frac{{L_{c} }}{L}$ at $\frac{l}{h} = 1$

Fig. 13

Variation of fundamental natural frequency of the cracked microbeam versus crack depth ratio $\eta = a/h$ for different values of dimensionless material length scale parameter $l/h$ at crack location $\frac{{L_{c} }}{L} = 0.5$

Fig. 14

Variation of fundamental natural frequency of the cracked microbeam versus crack location $\frac{{L_{c} }}{L}$ for different values of dimensionless material length scale parameter $l/h$ at crack depth ratio $\eta = 0.2$

Fig. 15

Variation of second natural frequency of the cracked microbeam versus crack location $\frac{{L_{c} }}{L}$ for different values of dimensionless material length scale parameter $l/h$ at crack depth ratio $\eta = 0.2$