An Approximate Analytical Solution
of Transversal Oscillations with Quintic
Nonlinearities
Nicolae Herisanu, Vasile Marinca, and Cristina Chilibaru-Opritescu
Abstract Free damped vibrations of a hinged–hinged Euler–Bernoulli beam subject
to a constant axial force at its free end is investigated. The quintic nonlinear equation
of motion is derived from Hamilton’s principle and then solved using the optimal
auxiliary function method (OAFM). Our proposed procedure is highly efficient and
controls the convergence of the solutions, ensuring an excellent accuracy after the
first iteration. Numerical values are also obtained in order to validate the analytical
results.
1 Introduction
The nonlinear oscillations of the beams received considerable attention over many
years. Structures like airplane wings, bridges, buildings, helicopter rotor blades, robot
arms, or drill strings can be modeled as a beam-like member. It is very important
to give an accurate analysis of the nonlinear vibration of these structures. Nonlinear
dynamic problems have fascinated engineers, mathematicians, and physicists for
a long time. Geometrically, nonlinear vibration of beams with different boundary
conditions has long history. Many analytical and numerical studies were reported in
the open literature.
Pakdemirli [1] showed that first discretizing the partial differential equation and
then applying perturbation methods lead to incorrect results. An accurate solution
is obtained by perturbation methods directly to the partial differential equations.
Approximate method for analyzing the vibrations of an Euler–Bernoulli beam resting
on a nonlinear elastic foundation is discussed by Nayfeh and Lacarbonara [2] in the
cases of primary and subharmonic resonance. They proved that the approximate
N. Herisanu (B) · C. Chilibaru-Opritescu
Politehnica University Timisoara, 300222 Timisoara, Romania
e-mail: nicolae.herisanu@upt.ro
N. Herisanu · V. Marinca
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_4
41
of Transversal Oscillations with Quintic
Nonlinearities
Nicolae Herisanu, Vasile Marinca, and Cristina Chilibaru-Opritescu
Abstract Free damped vibrations of a hinged–hinged Euler–Bernoulli beam subject
to a constant axial force at its free end is investigated. The quintic nonlinear equation
of motion is derived from Hamilton’s principle and then solved using the optimal
auxiliary function method (OAFM). Our proposed procedure is highly efficient and
controls the convergence of the solutions, ensuring an excellent accuracy after the
first iteration. Numerical values are also obtained in order to validate the analytical
results.
1 Introduction
The nonlinear oscillations of the beams received considerable attention over many
years. Structures like airplane wings, bridges, buildings, helicopter rotor blades, robot
arms, or drill strings can be modeled as a beam-like member. It is very important
to give an accurate analysis of the nonlinear vibration of these structures. Nonlinear
dynamic problems have fascinated engineers, mathematicians, and physicists for
a long time. Geometrically, nonlinear vibration of beams with different boundary
conditions has long history. Many analytical and numerical studies were reported in
the open literature.
Pakdemirli [1] showed that first discretizing the partial differential equation and
then applying perturbation methods lead to incorrect results. An accurate solution
is obtained by perturbation methods directly to the partial differential equations.
Approximate method for analyzing the vibrations of an Euler–Bernoulli beam resting
on a nonlinear elastic foundation is discussed by Nayfeh and Lacarbonara [2] in the
cases of primary and subharmonic resonance. They proved that the approximate
N. Herisanu (B) · C. Chilibaru-Opritescu
Politehnica University Timisoara, 300222 Timisoara, Romania
e-mail: nicolae.herisanu@upt.ro
N. Herisanu · V. Marinca
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_4
41