Proceedings of the Institution of Mechanical Engineers,Part K: Journal of Multi-body Dynamics

In this article, a mathematical model has been developed to predict the non-linear dynamic behaviour of a high-speed unbalanced rotating shaft due to ball size variations and varying number of balls. In the mathematical formulation, the contact between balls and inner/outer races is considered as a non-linear spring, whose stiffness is obtained using Hertzian elastic deformation theory. A detailed contact-damping model reflecting the influences of the surface profiles and the speeds of both contacting elements is developed and applied in the rolling element bearing formulation. This formulation also accounts for sources of non-linearity such as Hertzian contact force, varying number of balls, and off-sized balls resulting transition from non-contact to contact state between balls and races. The equations of motion of a rolling element bearing are formulated using Lagrange's equation considering the vibration characteristics of individual constituents such as inner race, outer race, rolling elements, and shaft. To overcome the strong non-linearity of the governing equations of motion, a modified Newmark-β time integration technique has been used to solve the equations of motion numerically. All results have been presented in the form of fast Fourier transformations and Poincaré maps. The highest radial vibrations due to ball size variation are at a speed of the number of balls times the cage speed (ω=kω_cage Hz). The other vibrations due to ball size variation also occurr at (X±k ω_cage), where k is a constant. When the number of balls is increased, the centre of oscillations approaches zero, implying a stiffer system. From this it can be predicted that increasing the number of balls will reduce the effect of the BPF. Hence it is implied from the analysis that the number of balls and the size variation of balls are important parameters and should be considered at the design stage.

In a previous investigation, a non-linear finite element formulation for modelling rail structural flexibility in multibody railroad vehicle systems was presented. This formulation required the use of isoparametric interpolation based on the finite element absolute nodal coordinate formulation (ANCF) in order to have a consistent interpolation for the geometry in the current deformed configuration. That is, the same interpolation functions used for the geometry description in the undeformed reference state are also used for predicting the geometry in the current deformed state. This approach, however, becomes less attractive when a degree of derivative continuity higher than the one provided by ANCF elements is required. In this investigation, an alternate simpler approach is proposed for modelling rail and track flexibility. The proposed new method, which is based on the floating frame of reference formulation, allows for general modelling of rail and track flexibility, can be used to impose a higher degree of continuity on the derivatives at the nodes that define the rail geometry in the reference configuration, and can be used to systematically couple finite element computer programs with flexible multibody system codes, allowing for the development of detailed track models that include rail, tie, and fastener flexibility as well as soil characteristics. In the proposed method, track geometry is defined in a preprocessor computer program using nodal points. In addition to the location of the nodes and Euler angles that define the orientation of the profile frame at the nodes, first, second, and third derivatives of the locations and Euler angles are defined by the preprocessor computer program. Numerical examples are presented in order to demonstrate the effect of track deformation in railroad applications.

This article discusses one of the solutions proposed in the literature to Gauss’ principle known as principle of ‘least constraint’, proposing a clarifying interpretation that allows significant computational improvements when considering its application to problems with non-diagonal mass matrix. The case of non-symmetric mass matrix is briefly discussed as well.

This article presents the modelling principles of the joint constraints for flexible multibody systems. The joints are composed using three basic constraint primitives which are derived including their first and second time derivatives as well as the components of the Jacobian matrix. The description of the derived components of constraint primitives can be used to develop a library of kinematic joints to use in multibody codes. In this study, the equations of motion are defined using generalized Newton—Euler equations where the deformations are accounted for by using the floating frame of reference formulation with modal coordinates. The deformation modes used in the floating frame of reference formulation are obtained from the finite-element analysis by employing the lumped mass matrix. Dynamic analysis of a mechanism consisting of rigid and flexible bodies is used to illustrate the validity of the constraint formulation.

One of the characteristics of floating frame of reference (FFR) formulations for flexible multibody dynamics is the fact that the inertia terms are highly non-linear. At every time-step, both the mass matrix and the velocity-dependent forces vector must be updated, and this can become the most CPU intensive task. This work studies the efficiency of two different methods for performing this operation, when applied to both a formulation in absolute coordinates and another in relative coordinates.

The first method calculates the inertia terms by projecting the finite element (FE) mass matrix into the generalized coordinates, by means of a variable projection matrix. The second one calculates the inertia shape integrals at a preprocessing stage and uses them for obtaining the inertia terms in a more efficient way, at the cost of a more involved implementation. Both methods have been tested when used in combination with either the FFR absolute or relative formulation, by simulating a vehicle with 12 flexible elements. The results show that the performance can be considerably increased by means of the preprocessing method, especially in the case of large FE models, whereas, for small models, the projection method can be more convenient due to its simplicity.