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Microsystem Technologies

, Volume 24, Issue 7, pp 3217–3223 | Cite as

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

  • Abbas Rahi
Correction
  • 187 Downloads

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 \( \eta = 0.2 \)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mechanical and Energy EngineeringShahid Beheshti University, A.C.TehranIran

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