A comparative study of computer models for friction and their influence on dynamics of the heavy rigid body on a horizontal surface

Using an example of a heavy rigid body moving on a horizontal surface and having with it a permanent contact the process of construction and verification for spatial dynamical models of the multibody systems is analyzed. Two approaches to formal representation of the models: object-oriented, and bond graph based are applied. Energy based similarities between these approaches are analyzed. A detailed description of the bond graph representation for the most general type of constraint is presented. It turned out the resulting total bond graph model of the multibody system dynamics always has exactly a so-called canonical junction structure. This representation has a tight correspondence with our object-oriented implementation of the mechanical constraint architecture. As an example Modelica implementation of several classes in the row for mechanical contact is investigated. Computer implementations for three examples of the heavy rigid body dynamics are under investigation: (a) the rattleback, (b) example of A. P. Markeev, (c) the Tippe- Top. Among all of three examples each one demonstrates in its own manner a peculiar dynamical behaviour

How One Can Arrange the Multibody System Dynamics Computer Model

The symplest computer models for mechanical systems can be organized using Cauchy normal form. Further complicating the mechanical model is able to bring us to application the implicit functions of next complexity level. Process of the multibody system computer models development is of special difficulty. There exist different ways for organizing such a models. Mainly these ways are reduced to the model transformation to the form of differential-algebraic equations (DAEs). These latter ones correspond to Lagrange equations of the first kind. Note that usually differential equations of DAEs mentioned correspond to dynamical and kinematical equations of mechanics, while the algebraic equations are generated by constraints. Computational experience makes it possible to classify objects of the multibody system dynamics [1]. Such a model includes two classes of objects. They correspond to notions of 'body' and 'constraint'. Let us also remark that these two classes of objects define the structure of the undirected graph such that 'bodies' play a role of the graph vertices, while 'constraints' play the role of edges. There exists yet another graph interpretation using the bi-chromatic bipartite graph. In this case both bodies and constraints are interpreted as vertices. Objects of bodies compose a partition and are coloured by one colour while objects of constraints compose the complement partition of the whole graph and are coloured by another colour. Edges connecting vertices of partitions for the graph are arranged in a way such that for any vertex of constraint there exist exactly two vertices of class 'body' thus implementing participation in the constraint. Two ways for the multibody system dynamics computer model graph composition mentioned above define ways for constructing the visual model of such a system thus defining ports interconnection structure. Different cases of the multibody system dynamics computer model implementation were analysed as an examples. Models under construction are the following ones: (a) Rattleback; (b) Snakeboard; (c) Skateboard; (d) Tippe-Top; (e) Ball Bearing; (f) Spur Involute Gear; (g) Omni­Vehicle

Mechanical Constraint Arrangement and Its Multibond Graph Representation

When developing a computer model of the multibody system (MBS) dynamics it is interesting to have a unified technology to construct the models in an efficient way. It turns out object-oriented approach provides a tools to resolve such a problem successively step by step. One of these unified ways is connected tightly with the so-called multiport representation of the models initially based on the bond graph use. These latter ones in turn based on the idea of energy exchanges, and substantially on energy conservation for physically interconnected subsystems of any engineering type. A detailed description of the multibond graph representation for the most general type of constraint is presented. It turned out the resulting total multibond graph model of the multibody system dynamics always has exactly a canonical junction structure. This representation has a tight correspondence with our previous object-oriented implementation of the mechanical constraint architecture. Computational experience makes it possible to classify objects of the multibody system dynamics. Such a model includes two classes of objects. They correspond to notions of 'body' and 'constraint'. Each of these notions indeed corresponds to the certain type of the multibond graph junction

How one can introduce compliance into computer models of the multibody dynamics using features of object-oriented modeling

Dynamics of a multibody system is simulated in a most universal way in case of contacts having a compliance property. This latter case is implemented usually by elasticity / viscosity along the direction normal to the rigid body outer / inner surface and by friction along its tangent direction. The Hertz model is one of the most popular elastic contact models for engineering applications. Object-oriented approach for building up the multibody dynam-ics model simulating compliant contacts is under development in this paper. A technology for constructing classes-templates is applied to build up contact objects in the dynamical model. The Hertz contact model is under consideration as a simplest example

Model of the tethered space system in vicinity of ellipsoidal asteroid and its approximations

While planning missions in vicinity of an asteroid/comet body one has to take into account several dynamical problems to overcome and among them are: (a) irregular distribution of the body internal masses; (b) too weak gravity acceleration near the body surface. In case of (a) one offers to apply approximate models of gravity. As an example we consider the case of a triaxial ellipsoid. For the problem (b) we apply docking procedures with help of anchor and a connecting tether. For computing the force field of gravity being generated by the ellipsoid of three axes one has to calculate several values of elliptic integrals at each instant of the simulation process. For this we apply original algorithm interpreting elliptic integrals as a state variables in additional to dynamics system of ODEs. To resolve the problem (b) we use so-called hybrid automata to build up the tethered interconnection between a spacecraft and the asteroid. Ellipsoidal asteroid performs free rotary motions about its mass center thus performing the Euler case of the rigid body rotary motion. The spacecraft moves under the force of gravity from the asteroid and under the tether tension, in case of the constraint being imposed. So we have so-called restricted dynamical model because the asteroid does not “feel” any force from the spacecraft. In addition to the hybrid automata dynamical model including impacts on constraint we also consider approximations of this model being really regularizations of the impact process. All these models are analysed and compared numerically

Contact types hierarchy and its object-oriented implementation

Technology of the object-oriented implementation for the multibody dynamics models is the key feature when developing the corresponding computer structures. We are based on an approach originating from concepts explained earlier. Following the guidelines outlined there one can develop the family of the constraint abstractions being adapted to any type of the machinery applications and relatively easily implement corresponding family of Modelica models. One also can reorder these classes hierarchically using sequences of the behaviour inheritance. Solutions concerning contact problems and corresponding examples are under consideration

A comparative study of computer models for friction and their influence on dynamics of the heavy rigid body on a horizontal surface

Using an example of a heavy rigid body moving on a horizontal surface and having with it a permanent contact the process of construction and verification for spatial dynamical models of the multibody systems is analyzed. Two approaches to formal representation of the models: object-oriented, and bond graph based are applied. Energy based similarities between these approaches are analyzed. A detailed description of the bond graph representation for the most general type of constraint is presented. It turned out the resulting total bond graph model of the multibody system dynamics always has exactly a so-called canonical junction structure. This representation has a tight correspondence with our object-oriented implementation of the mechanical constraint architecture. As an example Modelica implementation of several classes in the row for mechanical contact is investigated. Computer implementations for three examples of the heavy rigid body dynamics are under investigation: (a) the rattleback, (b) example of A. P. Markeev, (c) the Tippe- Top. Among all of three examples each one demonstrates in its own manner a peculiar dynamical behaviour

Revised and Improved Implementation of the Spur Involute Gear Dynamical Model

Abstract An improved model having new, more realistic, properties is constructed with use of previously implemented approach for building up a model of the spur involute gear dynamics. First of all, an algorithm for contact tracking of cylindrical surfaces directed by involutes was rearranged. This algorithm is "simply" reduced to tracking the two involutes. A result is that common line normal to these contact curves always coincides with the line of action. This property permits obtaining direct simple formulae for contact computations. A backlash in gearbox is also taken into account in the model under consideration. This means that a loss of contact between the teeth is possible as gearwheels rotate. This may then cause an appearance of a contact patch during the reversal. Furthermore, a dynamical reasons may force the mesh process to return to the former mode of the forward stroke and so fourth. All such scenarios for switching modes are implemented in the model in a unified way. A time overlapping of contacts between teeth pairs is used to ensure the mesh reliability. This property is also implemented in the described dynamical model. New contact of the next pair of teeth arises and starts its motion along the line of action before the old contact leaves this line at the point of teeth disengagement

To the 70th anniversary of Ivan Korshykov

The article highlights the live milestones of the famous Ukrainian biologist Ivan Korshykov