### Abstract

A mathematical model of the loss of dynamic stability of curvilinear size-dependent MEMS and NEMS elements embedded in a temperature field and subjected to large deflections was derived and studied. The fundamental governing dynamical equations of MEMS/NEMS members were yielded by Hamilton's principle. The investigations were based on combining the modified couple stress theory, the first-order approximation kinematic (Euler–Bernoulli) model, the von Kármán geometric non-linearity, and the Duhamel–Neumann law regarding the temperature input (the beam material is elastic, isotropic and there are no constraints imposed on the temperature distribution). The temperature field was defined by solving a heat transfer equation. The computational algorithm was based on the finite difference method and the Runge–Kutta method. The numerical methods were validated by estimating the temporal and spatial convergence and reliability of the obtained solution was validated with the Lyapunov exponents obtained by qualitatively different methods. A few case studies related to the loss of stability, the magnitude of the size-dependent parameter, the type and intensity of the temperature input, and the parameters of uniformly distributed transverse load were investigated.

Original language | English |
---|---|

Pages (from-to) | 283-296 |

Number of pages | 14 |

Journal | Applied Mathematical Modelling |

Volume | 67 |

DOIs | |

Publication status | Published - 1 Mar 2019 |

### Fingerprint

### Keywords

- Beams and columns
- Dynamics
- Finite differences
- Microstructures
- Thermal stress

### ASJC Scopus subject areas

- Modelling and Simulation
- Applied Mathematics

### Cite this

*Applied Mathematical Modelling*,

*67*, 283-296. https://doi.org/10.1016/j.apm.2018.10.026

**Size-dependent non-linear dynamics of curvilinear flexible beams in a temperature field.** / Krysko, V. A.; Awrejcewicz, J.; Kutepov, I. E.; Babenkova, T. V.; Krysko, A. V.

Research output: Contribution to journal › Article

*Applied Mathematical Modelling*, vol. 67, pp. 283-296. https://doi.org/10.1016/j.apm.2018.10.026

}

TY - JOUR

T1 - Size-dependent non-linear dynamics of curvilinear flexible beams in a temperature field

AU - Krysko, V. A.

AU - Awrejcewicz, J.

AU - Kutepov, I. E.

AU - Babenkova, T. V.

AU - Krysko, A. V.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - A mathematical model of the loss of dynamic stability of curvilinear size-dependent MEMS and NEMS elements embedded in a temperature field and subjected to large deflections was derived and studied. The fundamental governing dynamical equations of MEMS/NEMS members were yielded by Hamilton's principle. The investigations were based on combining the modified couple stress theory, the first-order approximation kinematic (Euler–Bernoulli) model, the von Kármán geometric non-linearity, and the Duhamel–Neumann law regarding the temperature input (the beam material is elastic, isotropic and there are no constraints imposed on the temperature distribution). The temperature field was defined by solving a heat transfer equation. The computational algorithm was based on the finite difference method and the Runge–Kutta method. The numerical methods were validated by estimating the temporal and spatial convergence and reliability of the obtained solution was validated with the Lyapunov exponents obtained by qualitatively different methods. A few case studies related to the loss of stability, the magnitude of the size-dependent parameter, the type and intensity of the temperature input, and the parameters of uniformly distributed transverse load were investigated.

AB - A mathematical model of the loss of dynamic stability of curvilinear size-dependent MEMS and NEMS elements embedded in a temperature field and subjected to large deflections was derived and studied. The fundamental governing dynamical equations of MEMS/NEMS members were yielded by Hamilton's principle. The investigations were based on combining the modified couple stress theory, the first-order approximation kinematic (Euler–Bernoulli) model, the von Kármán geometric non-linearity, and the Duhamel–Neumann law regarding the temperature input (the beam material is elastic, isotropic and there are no constraints imposed on the temperature distribution). The temperature field was defined by solving a heat transfer equation. The computational algorithm was based on the finite difference method and the Runge–Kutta method. The numerical methods were validated by estimating the temporal and spatial convergence and reliability of the obtained solution was validated with the Lyapunov exponents obtained by qualitatively different methods. A few case studies related to the loss of stability, the magnitude of the size-dependent parameter, the type and intensity of the temperature input, and the parameters of uniformly distributed transverse load were investigated.

KW - Beams and columns

KW - Dynamics

KW - Finite differences

KW - Microstructures

KW - Thermal stress

UR - http://www.scopus.com/inward/record.url?scp=85055907562&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055907562&partnerID=8YFLogxK

U2 - 10.1016/j.apm.2018.10.026

DO - 10.1016/j.apm.2018.10.026

M3 - Article

AN - SCOPUS:85055907562

VL - 67

SP - 283

EP - 296

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

ER -