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It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton-Euler's approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical resolution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as function of the Moore-Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian and the coordinate partitioning method.

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Lateral coordinate of the wheel surface xL Length of the strip x,y,z Cartesian coordinates y0 One dimension of the SDEC α Direction of the linear creepage γ Tangent angle of the cross-section γ Right-hand side of the acceleration constraint equations vector δ Penetration magnitude max  Maximum penetration velocity ΔFx Deviation of the longitudinal creep force ΔFy Deviation of the lateral creep force ΔMz Deviation of the spin creep moment Δr Step size for the radial coordinate Δs Width of the strip Δθ Step size for the angular coordinate ε Parameter that takes into account the existing deformation η Normalized lateral creepage θ Angular coordinate κ Curvature λ Lagrange multipliers vector μ Friction coefficient ν Magnitude of the linear creepages ξ Normalized longitudinal creepage σ Poisson ratio υx Longitudinal creepage υy Lateral creepage φ Spin creepage Φq Jacobian matrix of the constraint equations χ Normalized spin creepage ψ Shape factor of SDEC ω Angular velocity vector

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In railway dynamics, the interpolation of lookup tables (LUTs) is a procedure to reduce the computational effort when computing the wheel-rail interaction forces. However, the generation of LUTs with multiple inputs and multiple outputs is a challenging task for which issues such as their minimal size and uniform accuracy over the LUT domain have not been systematically addressed before. This work presents a comprehensive methodology for a detailed analysis of general LUTs, identifying ways to improve them. First, an analysis of the variation of the input parameters is made and the interpolation error is assessed on the cells and edges of the original table. From this analysis, two enhanced LUTs are proposed. One is approximately 5 times smaller than the original but holds similar accuracy. The other table exhibits half of the maximum interpolation error of the original LUT but holds an identical size. This methodology is demonstrated here using the recently published Kalker Book of Tables for Non-Hertzian contact (KBTNH) but it can be used by any other LUT approach in order to improve accuracy and/or to reduce size.

The wheel-rail contact modelling problem assumes a preponderant role on the dynamic analysis of railway systems using multibody systems formulations. The accurate and efficient evaluation of both location and magnitude of the wheel-rail contact forces is fundamental for the development of reliable computational tools. The wheel concave zone might be a source of numerical difficulties when searching the contact points, which has been neglected in several works. Here, it is demonstrated that the minimum distance method does not always converge when the wheel surface is not fully convex, being an alternative methodology proposed to perform the contact detection. This approach examines independently the contact between each wheel strip and the rail, where the maximum virtual penetration is determined and associated with the location of the contact point. Then, an Hertzian-based force model is considered for both normal and tangential forces. The results obtained from dynamic simulations show that the minimum distance method and the proposed methodology provide a similar response for simplified wheel profiles. However, the new approach described here is reliable in the identification of the contact point when realistic wheel profiles are considered, which is not the case with the minimum distance method.

An investigation on the dynamic modeling and analysis of spatial mechanisms with spherical clearance joints including friction is presented. For this purpose, the ball and the socket, which compose a spherical joint, are modeled as two individual colliding components. The normal contact-impact forces that develop at the spherical clearance joint are determined by using a continuous force model. A continuous analysis approach is used here with a Hertzian-based contact force model, which includes a dissipative term representing the energy dissipation during the contact process. The pseudopenetration that occurs between the potential contact points of the ball and the socket surface, as well as the indentation rate play a crucial role in the evaluation of the normal contact forces. In addition, several different friction force models based on the Coulomb's law are revisited in this work. The friction models utilized here can accommodate the various friction regimens and phenomena that take place at the contact interface between the ball and the socket. Both the normal and tangential contact forces are evaluated and included into the systems' dynamics equation of motion, developed under the framework of multibody systems formulations. A spatial four-bar mechanism, which includes a spherical joint with clearance, is used as an application example to examine and quantify the effects of various friction force models, clearance sizes, and the friction coefficients.