Modelling of Frictional Contacts in 3D
Dynamics of a Rigid Body
Jan Awrejcewicz and Grzegorz Kudra
Abstract There is considered a system of a spatial double pendulum with rigid
movable obstacle, consisting of two links connected to each other and suspended
on a shaft performing rotational motion about its horizontal axis according to a
given function of time (kinematic driving). The links are connected by the use of
two universal joints. The second link ends with a ball which can come into contact
(impacts and permanent contact) with a planar and rotating obstacle situated below
the pendulum. There is presented mathematical model of dynamics based on the
Lagrange formulation. In this work, we use and expand our earlier developed models
of contact forces (resulting friction force and rolling resistance). The friction models
are based on the integral model developed assuming developed sliding on a planar
contact area, where at each point, the classical Coulomb’s friction law is valid. The
integral models are then replaced by special approximations being more suitable for
fast numerical simulations. In the present work, we model impacts with non-point
frictional contacts assuming Hertzian compliance of the obstacle. The constructed
models of 3D dynamics of a rigid body and the planned experimental investigations
allow us to perform the tests of importance of the particular individual elements
of the models and may lead to general conclusions about modelling and effective
computer simulations of mechanical systems with 3D frictional contacts. We report
bifurcation dynamics using bifurcation diagrams, Poincaré sections as well as the
largest Lyapunov exponent.
1 Introduction
Pendulum-based mechanical systems serve as a paradigmatic model for analysis
of many problems in nonlinear dynamics, mechanical engineering, biomechanics,
control theory and mechatronics. Among different mechanical systems, one can
encounter models based on spherical pendulum [1]. For example, in the work [2], the
J. Awrejcewicz (B) · G. Kudra
Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
e-mail: jan.awrejcewicz@p.lodz.pl
© 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_1
3
Dynamics of a Rigid Body
Jan Awrejcewicz and Grzegorz Kudra
Abstract There is considered a system of a spatial double pendulum with rigid
movable obstacle, consisting of two links connected to each other and suspended
on a shaft performing rotational motion about its horizontal axis according to a
given function of time (kinematic driving). The links are connected by the use of
two universal joints. The second link ends with a ball which can come into contact
(impacts and permanent contact) with a planar and rotating obstacle situated below
the pendulum. There is presented mathematical model of dynamics based on the
Lagrange formulation. In this work, we use and expand our earlier developed models
of contact forces (resulting friction force and rolling resistance). The friction models
are based on the integral model developed assuming developed sliding on a planar
contact area, where at each point, the classical Coulomb’s friction law is valid. The
integral models are then replaced by special approximations being more suitable for
fast numerical simulations. In the present work, we model impacts with non-point
frictional contacts assuming Hertzian compliance of the obstacle. The constructed
models of 3D dynamics of a rigid body and the planned experimental investigations
allow us to perform the tests of importance of the particular individual elements
of the models and may lead to general conclusions about modelling and effective
computer simulations of mechanical systems with 3D frictional contacts. We report
bifurcation dynamics using bifurcation diagrams, Poincaré sections as well as the
largest Lyapunov exponent.
1 Introduction
Pendulum-based mechanical systems serve as a paradigmatic model for analysis
of many problems in nonlinear dynamics, mechanical engineering, biomechanics,
control theory and mechatronics. Among different mechanical systems, one can
encounter models based on spherical pendulum [1]. For example, in the work [2], the
J. Awrejcewicz (B) · G. Kudra
Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
e-mail: jan.awrejcewicz@p.lodz.pl
© 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_1
3