Study of the Vibrations of a System
Consisting in Cantilever Beams
Maria-Luiza Bes , liu-Gherghescu, Nicolae-Doru St˘ anescu , Nicolae Pandrea,
and Dinel Popa
Abstract We consider a system consisting in two cantilever beams jointed one to
another at the free ends. On the normal direction of the beams acts a force that
produces the deformation of the beams. We assume that these deformations have
great values, and we determine the equations which give the shape of the deformed
beams and the axial force that acts upon the system. By variation of the normal force,
one may obtain different shapes of beams and axial force. We present a numerical
example for which we have drawn the diagrams of variation of the characteristic
parameters.
1 Introduction
The biography discusses only the case of a single cantilever beam acted by different
concentrated or distributed forces and moments. Various methods were proposed for
the determination of the shape of the beam and its elongation at different points [1–
8]. Consideration about the elastic elements is presented in [9], while a method for
the elimination of the Lagrange multipliers is described in [10]. These methods were
presented in detail in [11], and all of them are valid in some hypotheses, especially
the length of the deformed beam is equal to the length of the non-deformed beam.
For small deformations of the cantilever beam, the general theory known from
the strength of materials can be applied. Large deformations appear for high values
of the forces and moments or in the cases of small values of the inertial parameters
or Young’s modulus. The last situation characterizes the beams obtained by additive
manufacturing (AM), especially by fused deposition material (FDM).
General situation for a system consisting in many cantilever beams was not
studied in the literature because of the complicate mathematics that intervenes in
the modelling, the hypothesis of constant lengths of the beams does not hold true. In
this paper, we consider a system of two cantilever beams linked one to another by
M.-L. Bes , liu-Gherghescu · N.-D. St˘ anescu (B) · N. Pandrea · D. Popa
University of Pitesti, Pitesti 110040, Romania
e-mail: s_doru@yahoo.com
© 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_6
59
Consisting in Cantilever Beams
Maria-Luiza Bes , liu-Gherghescu, Nicolae-Doru St˘ anescu , Nicolae Pandrea,
and Dinel Popa
Abstract We consider a system consisting in two cantilever beams jointed one to
another at the free ends. On the normal direction of the beams acts a force that
produces the deformation of the beams. We assume that these deformations have
great values, and we determine the equations which give the shape of the deformed
beams and the axial force that acts upon the system. By variation of the normal force,
one may obtain different shapes of beams and axial force. We present a numerical
example for which we have drawn the diagrams of variation of the characteristic
parameters.
1 Introduction
The biography discusses only the case of a single cantilever beam acted by different
concentrated or distributed forces and moments. Various methods were proposed for
the determination of the shape of the beam and its elongation at different points [1–
8]. Consideration about the elastic elements is presented in [9], while a method for
the elimination of the Lagrange multipliers is described in [10]. These methods were
presented in detail in [11], and all of them are valid in some hypotheses, especially
the length of the deformed beam is equal to the length of the non-deformed beam.
For small deformations of the cantilever beam, the general theory known from
the strength of materials can be applied. Large deformations appear for high values
of the forces and moments or in the cases of small values of the inertial parameters
or Young’s modulus. The last situation characterizes the beams obtained by additive
manufacturing (AM), especially by fused deposition material (FDM).
General situation for a system consisting in many cantilever beams was not
studied in the literature because of the complicate mathematics that intervenes in
the modelling, the hypothesis of constant lengths of the beams does not hold true. In
this paper, we consider a system of two cantilever beams linked one to another by
M.-L. Bes , liu-Gherghescu · N.-D. St˘ anescu (B) · N. Pandrea · D. Popa
University of Pitesti, Pitesti 110040, Romania
e-mail: s_doru@yahoo.com
© 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_6
59