A Reduced and Linearized High Fidelity Waveboard Multibody Model for Stability Analysis

2022 ◽  
Author(s):  
Alfonso García-Agúndez Blanco ◽  
Daniel García Vallejo ◽  
Emilio Freire ◽  
Aki Mikkola
Keyword(s):  
Multibody Systems ◽  
Jacobian Matrix ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Nonholonomic Constraints ◽  
Design Parameters ◽  
Constrained Multibody Systems ◽  
Multibody Model ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract In this paper, the stability of a waveboard, a human propelled two-wheeled vehicle consisting in two rotatable platforms, joined by a torsion bar and supported on two caster wheels, is analysed. A multibody model with holonomic and nonholonomic constraints is used to describe the system. The nonlinear equations of motion, which constitute a Differential-Algebraic system of equations (DAE system), are linearized along the steady forward motion resorting to a recently validated linearization procedure, which allows the maximum possible reduction of the linearized equations of motion of constrained multibody systems. The approach enables the generation of the Jacobian matrix in terms of the geometric and dynamic parameters of the multibody system, and the eigenvalues of the system are parameterized in terms of the design parameters. The resulting minimum set of linear equations leads to the elimination of spurious null eigenvalues, while retaining all the stability information in spite of the reduction of the Jacobian matrix. The linear stability results of the waveboard obtained in previous work are validated with this approach. The procedure shows an excellent computational efficiency with the waveboard, its utilization being highly advisable to linearize the equations of motion of complex constrained multibody systems.

Linear Stability Analysis of a Waveboard Multibody Model With a Minimal Set of Equations

2021 ◽  
Author(s):  
A. G. Agúndez ◽  
D. García-Vallejo ◽  
E. Freire ◽  
A. M. Mikkola
Keyword(s):  
Stability Analysis ◽  
Multibody System ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Design Parameters ◽  
Multibody Model ◽  
Aspect Ratios ◽  
Holonomic And Nonholonomic Constraints ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract In this paper, the stability of a waveboard, the skateboard consisting in two articulated platforms, coupled by a torsion bar and supported of two caster wheels, is analysed. The waveboard presents an interesting propelling mechanism, since the rider can achieve a forward motion by means of an oscillatory lateral motion of the platforms. The system is described using a multibody model with holonomic and nonholonomic constraints. To perform the stability analysis, the nonlinear equations of motion are linearized with respect to the forward upright motion with constant speed. The linearization is carried out resorting to a novel numerical linearization procedure, recently validated with a well-acknowledged bicycle benchmark, which allows the maximum possible reduction of the linearized equations of motion of multibody systems with holonomic and nonholonomic constraints. The approach allows the expression of the Jacobian matrix in terms of the main design parameters of the multibody system under study. This paper illustrates the use of this linearization approach with a complex multibody system as the waveboard. Furthermore, a sensitivity analysis of the eigenvalues considering different scenarios is performed, and the influence of the forward speed, the casters’ inclination angle and the tori aspect ratios of the toroidal wheels on the stability of the system is analysed.

Analysis of the Nonlinear Dynamics of a Railway Vehicle Wheelset

1973 ◽  
Vol 95 (1) ◽  
pp. 28-35 ◽  
Author(s):  
E. Harry Law ◽  
R. S. Brand
Keyword(s):  
Lateral Displacement ◽  
Equations Of Motion ◽  
Vertical Displacement ◽  
Linear Analysis ◽  
Design Parameters ◽  
Railway Vehicle ◽  
Nonlinear Variation ◽  
Constraint Equations ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The nonlinear equations of motion for a railway vehicle wheelset having curved wheel profiles and wheel-flange/rail contact are presented. The dependence of axle roll and vertical displacement on lateral displacement and yaw is formulated by two holonomic constraint equations. The method of Krylov-Bogoliubov is used to derive expressions for the amplitudes of stationary oscillations. A perturbation analysis is then used to derive conditions for the stability characteristics of the stationary oscillations. The expressions for the amplitude and the stability conditions are shown to have a simple geometrical interpretation which facilitates the evaluation of the effects of design parameters on the motion. It is shown that flange clearance and the nonlinear variation of axle roll with lateral displacement significantly influence the motion of the wheelset. Stationary oscillations may occur at forward speeds both below and above the critical speed at which a linear analysis predicts the onset of instability.

Symbolic Linearized Equations for Nonholonomic Multibody Systems With Closed-Loop Kinematics

2018 ◽  
Author(s):  
Márton Kuslits ◽  
Dieter Bestle
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Linearized Equations ◽  
Nonlinear Equations Of Motion ◽  
Loop Constraints

Multibody systems and associated equations of motion may be distinguished in many ways: holonomic and nonholonomic, linear and nonlinear, tree-structured and closed-loop kinematics, symbolic and numeric equations of motion. The present paper deals with a symbolic derivation of nonlinear equations of motion for nonholonomic multibody systems with closed-loop kinematics, where any generalized coordinates and velocities may be used for describing their kinematics. Loop constraints are taken into account by algebraic equations and Lagrange multipliers. The paper then focuses on the derivation of the corresponding linear equations of motion by eliminating the Lagrange multipliers and applying a computationally efficient symbolic linearization procedure. As demonstration example, a vehicle model with differential steering is used where validity of the approach is shown by comparing the behavior of the linearized equations with their nonlinear counterpart via simulations.

On the Use of the Canonical Equations of Motion for the Dynamic Analysis of Constrained Multibody Systems

1992 ◽  
Author(s):  
E. Bayo ◽  
J. M. Jimenez
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
First Order ◽  
Canonical Variables ◽  
Lagrange’S Multipliers

Abstract We investigate in this paper the different approaches that can be derived from the use of the Hamiltonian or canonical equations of motion for constrained mechanical systems with the intention of responding to the question of whether the use of these equations leads to more efficient and stable numerical algorithms than those coming from acceleration based formalisms. In this process, we propose a new penalty based canonical description of the equations of motion of constrained mechanical systems. This technique leads to a reduced set of first order ordinary differential equations in terms of the canonical variables with no Lagrange’s multipliers involved in the equations. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency. In addition, we examine the use of the canonical equations based on independent coordinates, and conclude that in this second case the use of the acceleration based formulation is more advantageous than the canonical counterpart.

Stability Analysis of High-Speed Railway Vehicle Based on Multibody Dynamics Analysis

2013 ◽  
Vol 392 ◽  
pp. 156-160
Author(s):  
Ju Seok Kang
Keyword(s):  
Stability Analysis ◽  
Multibody Dynamics ◽  
High Speed ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Design Parameters ◽  
Railway Vehicle ◽  
Dynamics Analysis ◽  
Contact Points ◽  
The Stability

Multibody dynamics analysis is advantageous in that it uses real dimensions and design parameters. In this study, the stability analysis of a railway vehicle based on multibody dynamics analysis is presented. The equations for the contact points and contact forces between the wheel and the rail are derived using a wheelset model. The dynamics equations of the wheelset are combined with the dynamics equations of the other parts of the railway vehicle, which are obtained by general multibody dynamics analysis. The equations of motion of the railway vehicle are linearized by using the perturbation method. The eigenvalues of these linear dynamics equations are calculated and the critical speed is found.

Dynamics of Constrained Multibody Systems

1984 ◽  
Vol 51 (4) ◽  
pp. 899-903 ◽  
Author(s):  
J. W. Kamman ◽  
R. L. Huston
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Governing Equations ◽  
Dynamical Equations ◽  
Closed Loops ◽  
Constrained Multibody Systems ◽  
Constraint Equations ◽  
Cable Dynamics ◽  
Hanging Chain ◽  
A Chain

A new automated procedure for obtaining and solving the governing equations of motion of constrained multibody systems is presented. The procedure is applicable when the constraints are either (a) geometrical (for example, “closed-loops”) or (b) kinematical (for example, specified motion). The procedure is based on a “zero eigenvalues theorem,” which provides an “orthogonal complement” array which in turn is used to contract the dynamical equations. This contraction, together with the constraint equations, forms a consistent set of governing equations. An advantage of this formulation is that constraining forces are automatically eliminated from the analysis. The method is applied with Kane’s equations—an especially convenient set of dynamical equations for multibody systems. Examples of a constrained hanging chain and a chain whose end has a prescribed motion are presented. Applications in robotics, cable dynamics, and biomechanics are suggested.

Free Vibration of High-Order Curved Sandwich Beams

2001 ◽  
Author(s):  
Yeoshua Frostig ◽  
Elena Bozhevolnaya
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Free Vibrations ◽  
High Order ◽  
Design Parameters ◽  
Sandwich Beams ◽  
Natural Boundary Conditions ◽  
Thin Beams ◽  
Sandwich Core

Abstract A high-order model for free vibrations of singly curved sandwich beams is presented. The model takes into account the transverse flexibility of the sandwich core while the faces of the sandwich are treated as thin beams. Linear equations of motion as well as the natural boundary conditions are derived. It is shown for the curved sandwich beams, that there exist four eigen modes. A numerical analysis of free vibration of the simply supported beams is carried out. Effects of design parameters of the sandwich constituents on the eigen modes and their appropriate frequencies are investigated.

The Hunting Behavior of Conventional Railway Trucks

1972 ◽  
Vol 94 (2) ◽  
pp. 752-761 ◽  
Author(s):  
N. K. Cooperrider
Keyword(s):  
Equations Of Motion ◽  
Linear Equations ◽  
Nonlinear Effects ◽  
Complex Model ◽  
Railway Vehicle ◽  
Hunting Behavior ◽  
Wheel Slip ◽  
Wheel Load ◽  
Hunting Motion ◽  
Nonlinear Equations Of Motion

Railway vehicles under certain conditions experience sustained lateral oscillations during which the wheel flanges bang from one rail to the other. It has been found that this behavior, called hunting, only occurs above certain critical forward velocities. Approximations to these critical velocities have been found from a stability analysis of the linear equations of motion for many different railway vehicle models. Hunting is characterized by violent motions that impose large loads on the vehicle and track, and bring several important nonlinear effects into play. This paper reports results of an analysis of nonlinear equations of motion written for two models of a railway truck. The influence of the nonlinear effects on stability is determined and the character of the hunting motion is investigated. One model represents a truck whose axle bearings are rigidly held in the truck frame while the truck frame is connected through a suspension system to a reference that moves along the track with constant velocity. The more complex model includes additional suspension elements between the axle bearings and truck frame. The effects of flange contact, wheel slip and Coulomb friction are described by nonlinear expressions. These results show the significant influence of flange contact on stability, and illustrate the effects of vehicle and track parameters such as rail adhesion, forward velocity, and wheel load on the forces and power dissipation at the wheel-rail interface.

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