4
J. Awrejcewicz and G. Kudra
authors analysed the problem of control of the spherical pendulum. In the paper [3],
there is presented analysis of 3D frictionless double pendulum rotating with constant
angular velocity about its vertical axis. Another example of a multi-pendulum system
in 3D space is investigated in the work [4]. Since in robotics, many tasks require
making and braking contact with different subjects, some group of works concerns
the problems of pendulum dynamics with impact and friction phenomena [5, 6].
Real natural and engineering objects with frictional 3D contacts very often cannot
be modelled using classical one-dimensional friction model. As examples, one can
indicate rolling bearings, billiard ball, Celtic stone or other contact phenomena
encountered for example in robotics. Exact numerical modelling and simulation
require in general space discretization methods and lead to high computational costs.
Looking for simplified but realistic models, Contensou proposed an integral model
of resultant friction force assuming fully developed sliding and Coulomb friction law
at each element of the contact area [7]. The integral model can be then approximated
by the use of special algebraic functions in order to make the simulation faster [8].
Some authors developed approximations of the friction force and moment assuming
special contact pressure distributions allowing to model rolling resistance [9, 10].
The approximated models were tested during modelling of the selected mechanical
systems: wobblestone, billiard ball and full solid ellipsoid of the revolution [11–13].
In multibody system dynamics, the impact phenomena can be modelled as the
so-called hard or soft impacts [14, 15]. However, the impacts are usually modelled as
phenomena occurring at a certain point. Even if friction torque is taken into account,
the coupling between friction force and moment is neglected.
In this work, we join the friction models developed and presented in the works
[10–13] with impact model based on Hertzian stiffness and special model of nonlinear
damping. The present work is continuation and extension of the conference papers
[16, 17].
2 Mathematical Model
In Fig. 1a, there is presented a physical concept of the investigated double spatial
pendulum, where one can observe the fixed frame F, three connected solids (the
body 0, the pendulums 1 and 2) and the obstacle 3 in the form of the disc performing
rotational motion. The solid 0 performs rotational motion with respect to the fixed
frame. The solid 0, pendulum 1 and 2 are connected by the use of two Cardan–Hook
joints. There are introduced the following coordinate systems: the fixed (global)
reference frame O 1 x yz, pendulum 1 fixed reference frame O 1 x 1 y 1 z 1 and pendulum
2 fixed coordinate system O 2 x 2 y 2 z 2 . It is assumed that the origins O 1 and O 2 of the
introduced coordinate systems lie in the centres of the corresponding Cardan–Hook
joints (intersections of their axes) and that the body 0 rotates about the axis z.
It is assumed that the initial position of the system corresponds to the reference
frames O 1 x 1 y 1 z 1 and O 1 x yz overlapping each other and the axes of the coordinate
system O 2 x 2 y 2 z 2 being parallel to the corresponding axes of the reference frame
J. Awrejcewicz and G. Kudra
authors analysed the problem of control of the spherical pendulum. In the paper [3],
there is presented analysis of 3D frictionless double pendulum rotating with constant
angular velocity about its vertical axis. Another example of a multi-pendulum system
in 3D space is investigated in the work [4]. Since in robotics, many tasks require
making and braking contact with different subjects, some group of works concerns
the problems of pendulum dynamics with impact and friction phenomena [5, 6].
Real natural and engineering objects with frictional 3D contacts very often cannot
be modelled using classical one-dimensional friction model. As examples, one can
indicate rolling bearings, billiard ball, Celtic stone or other contact phenomena
encountered for example in robotics. Exact numerical modelling and simulation
require in general space discretization methods and lead to high computational costs.
Looking for simplified but realistic models, Contensou proposed an integral model
of resultant friction force assuming fully developed sliding and Coulomb friction law
at each element of the contact area [7]. The integral model can be then approximated
by the use of special algebraic functions in order to make the simulation faster [8].
Some authors developed approximations of the friction force and moment assuming
special contact pressure distributions allowing to model rolling resistance [9, 10].
The approximated models were tested during modelling of the selected mechanical
systems: wobblestone, billiard ball and full solid ellipsoid of the revolution [11–13].
In multibody system dynamics, the impact phenomena can be modelled as the
so-called hard or soft impacts [14, 15]. However, the impacts are usually modelled as
phenomena occurring at a certain point. Even if friction torque is taken into account,
the coupling between friction force and moment is neglected.
In this work, we join the friction models developed and presented in the works
[10–13] with impact model based on Hertzian stiffness and special model of nonlinear
damping. The present work is continuation and extension of the conference papers
[16, 17].
2 Mathematical Model
In Fig. 1a, there is presented a physical concept of the investigated double spatial
pendulum, where one can observe the fixed frame F, three connected solids (the
body 0, the pendulums 1 and 2) and the obstacle 3 in the form of the disc performing
rotational motion. The solid 0 performs rotational motion with respect to the fixed
frame. The solid 0, pendulum 1 and 2 are connected by the use of two Cardan–Hook
joints. There are introduced the following coordinate systems: the fixed (global)
reference frame O 1 x yz, pendulum 1 fixed reference frame O 1 x 1 y 1 z 1 and pendulum
2 fixed coordinate system O 2 x 2 y 2 z 2 . It is assumed that the origins O 1 and O 2 of the
introduced coordinate systems lie in the centres of the corresponding Cardan–Hook
joints (intersections of their axes) and that the body 0 rotates about the axis z.
It is assumed that the initial position of the system corresponds to the reference
frames O 1 x 1 y 1 z 1 and O 1 x yz overlapping each other and the axes of the coordinate
system O 2 x 2 y 2 z 2 being parallel to the corresponding axes of the reference frame