Optimal Auxiliary Functions Method
for Nonlinear Vibration of Doubly
Clamped Nanobeam Incorporating
the Casimir Force
Vasile Marinca and Nicolae Herisanu
Abstract The optimal auxiliary functions method is implemented in this paper to
derive an analytical approximate solution for the problem of nonlinear vibrations
of doubly clamped nanobeam incorporating the Casimir force. A closed form solution is obtained. This model accounts for the inherent nonlinearity of the Casimir
force and the geometric nonlinearity of mid-plane stretching. Our technique is optimized to accelerate the convergence of approximate solutions via the presence of
the optimal convergence-control parameters. The procedure does not depend upon
small or large parameters ensuring a very fast convergence, after the first iteration.
Numerical developments are presented in order to validate the analytical approximate
results.
1 Introduction
It is known that analytical approximate solutions of nonlinear differential equations
which describe various phenomena in science and technology are often difficult
to obtain. Micromechanical systems (MEMS), micro-sensors and actuators are an
extension of microelectronics and integrated circuits technology. Because of their
small size, low fabrication cost, performace and suitability for integration into
complex functional engineering systems, they are important components of many
commercial systems including metal alloy, accelerometers for airbag deployment in
automobiles, single crystal silicon switches, filters, resonators, sensors, ink jet printer
heads, atomic force microscopes, and so on. Nonlinearity usually arises from sources
such as electrostatic actuation itself, squeeze-film damping, geometric nonlinearities
and intermolecular forces, i.e., Casimir or van der Waals.
N. Herisanu (B)
Politehnica University Timisoara, 300222 Timisoara, Romania
e-mail: nicolae.herisanu@upt.ro
V. Marinca · N. Herisanu
Centre for Advanced Technical Research-CCTFA, Romanian Academy, Branch of Timisoara,
300222 Timisoara, Romania
© Springer Nature Switzerland AG 2021
N. Herisanu and V. Marinca (eds.), Acoustics and Vibration of Mechanical
Structures—AVMS 2019, Springer Proceedings in Physics 251,
https://doi.org/10.1007/978-3-030-54136-1_5
51
for Nonlinear Vibration of Doubly
Clamped Nanobeam Incorporating
the Casimir Force
Vasile Marinca and Nicolae Herisanu
Abstract The optimal auxiliary functions method is implemented in this paper to
derive an analytical approximate solution for the problem of nonlinear vibrations
of doubly clamped nanobeam incorporating the Casimir force. A closed form solution is obtained. This model accounts for the inherent nonlinearity of the Casimir
force and the geometric nonlinearity of mid-plane stretching. Our technique is optimized to accelerate the convergence of approximate solutions via the presence of
the optimal convergence-control parameters. The procedure does not depend upon
small or large parameters ensuring a very fast convergence, after the first iteration.
Numerical developments are presented in order to validate the analytical approximate
results.
1 Introduction
It is known that analytical approximate solutions of nonlinear differential equations
which describe various phenomena in science and technology are often difficult
to obtain. Micromechanical systems (MEMS), micro-sensors and actuators are an
extension of microelectronics and integrated circuits technology. Because of their
small size, low fabrication cost, performace and suitability for integration into
complex functional engineering systems, they are important components of many
commercial systems including metal alloy, accelerometers for airbag deployment in
automobiles, single crystal silicon switches, filters, resonators, sensors, ink jet printer
heads, atomic force microscopes, and so on. Nonlinearity usually arises from sources
such as electrostatic actuation itself, squeeze-film damping, geometric nonlinearities
and intermolecular forces, i.e., Casimir or van der Waals.
N. Herisanu (B)
Politehnica University Timisoara, 300222 Timisoara, Romania
e-mail: nicolae.herisanu@upt.ro
V. Marinca · N. Herisanu
Centre for Advanced Technical Research-CCTFA, Romanian Academy, Branch of Timisoara,
300222 Timisoara, Romania
© Springer Nature Switzerland AG 2021
N. Herisanu and V. Marinca (eds.), Acoustics and Vibration of Mechanical
Structures—AVMS 2019, Springer Proceedings in Physics 251,
https://doi.org/10.1007/978-3-030-54136-1_5
51