Vibrations with Collisions
of a Mechanical System with Elastic
Elements
Ionut , -Bogdan Dragna, Nicolae Pandrea, Nicolae-Doru St˘ anescu ,
and Dinel Popa
Abstract In this paper, we present a mechanical system consisting in an arbitrary
number of masses linked one to another by springs. The magnitudes of the vibrations
are limited by some retainers located on the directions of vibrations. One knows
the coefficients of restitution and the geometric and mechanical parameters of the
system. The first index 0 mass is excited by a harmonic type force. We determine
the equations of motion corresponding to each mass and the laws of motion, which,
due to the collisions, are nonlinear ones. We also describe a numerical example for
which we have drawn the diagrams of variation of the characteristic parameters.
1 Introduction
The modern approach of collisions is based on the theory of screws and uses the
concept of inertance introduced in [22, 23]. The references [1–12, 14–17, 19–21,
24–36] discuss different aspects of collisions between two rigid bodies or between a
rigid body and an obstacle considering or not the existence of friction at the contact
point. The collisions are studied based on the hypotheses stated in [6, 22, 23], that is
• the impact forces are so high that one may neglect the other forces in system;
• collision consists in two phases: compression and expansion;
• positions remain constant during the shock;
• tangential stiffness is infinite.
For collisions without friction, one may prove [6, 22, 23] that the Newton, Poisson
and energetic coefficients of restitution are equal and have values in interval [0, 1].
For collisions with friction, this equality between the coefficients of restitution is no
longer valid, and there are examples [23] in which the coefficients of restitution may
exceed 1. In addition, complex phenomena like jamb [3] may occur. It is also proved
the existence of principal sliding directions [23].
I.-B. Dragna · N. Pandrea · N.-D. St˘ anescu (B) · 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_7
67
of a Mechanical System with Elastic
Elements
Ionut , -Bogdan Dragna, Nicolae Pandrea, Nicolae-Doru St˘ anescu ,
and Dinel Popa
Abstract In this paper, we present a mechanical system consisting in an arbitrary
number of masses linked one to another by springs. The magnitudes of the vibrations
are limited by some retainers located on the directions of vibrations. One knows
the coefficients of restitution and the geometric and mechanical parameters of the
system. The first index 0 mass is excited by a harmonic type force. We determine
the equations of motion corresponding to each mass and the laws of motion, which,
due to the collisions, are nonlinear ones. We also describe a numerical example for
which we have drawn the diagrams of variation of the characteristic parameters.
1 Introduction
The modern approach of collisions is based on the theory of screws and uses the
concept of inertance introduced in [22, 23]. The references [1–12, 14–17, 19–21,
24–36] discuss different aspects of collisions between two rigid bodies or between a
rigid body and an obstacle considering or not the existence of friction at the contact
point. The collisions are studied based on the hypotheses stated in [6, 22, 23], that is
• the impact forces are so high that one may neglect the other forces in system;
• collision consists in two phases: compression and expansion;
• positions remain constant during the shock;
• tangential stiffness is infinite.
For collisions without friction, one may prove [6, 22, 23] that the Newton, Poisson
and energetic coefficients of restitution are equal and have values in interval [0, 1].
For collisions with friction, this equality between the coefficients of restitution is no
longer valid, and there are examples [23] in which the coefficients of restitution may
exceed 1. In addition, complex phenomena like jamb [3] may occur. It is also proved
the existence of principal sliding directions [23].
I.-B. Dragna · N. Pandrea · N.-D. St˘ anescu (B) · 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_7
67