The Use of Zero-Mass Particles in Analytical and Multi-body Dynamics: sphere rolling on an arbitrary surface

2021 ◽  
pp. 1-24
Author(s):  
Firdaus Udwadia ◽  
Nami Mogharabin
Keyword(s):  
Equations Of Motion ◽  
Classical Problem ◽  
Analytical Dynamics ◽  
Homogeneous Sphere ◽  
Constrained Mechanical Systems ◽  
Zero Mass ◽  
Significant Difference ◽  
Mass Matrices ◽  
Body Dynamics ◽  
Multi Body

Abstract Zero-mass particles are, as a rule, never used in analytical dynamics, because they lead to singular mass matrices. However, recent advances in the development of the explicit equations of motion of constrained mechanical systems with singular mass matrices permit their use under certain circumstances. This paper shows that the use of such particles can be very efficacious in some problems in analytical dynamics that have resisted easy, general formulations, and in obtaining the equations of motion for complex multi-body systems. We explore the ease and simplicity that suitably used zero-mass particles can provide in formulating and simulating the equations of motion of a rigid, non-homogeneous sphere rolling under gravity, without slipping, on an arbitrarily prescribed surface. Computational results comparing the significant difference in the motion of a homogeneous sphere and a non-homogeneous sphere rolling down an asymmetric arbitrarily prescribed surface are obtained, along with measures of the accuracy of the computations. While the paper shows the usefulness of zero-mass particles applied to the classical problem of a rolling sphere, the development given is described in a general enough manner to be applicable to numerous other problems in analytical and multi-body dynamics that may have much greater complexity.

Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics

2006 ◽  
Vol 462 (2071) ◽  
pp. 2097-2117 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Explicit Form ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Constrained Motion ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
Mass Matrices ◽  
Body Dynamics ◽  
Multi Body ◽  
Explicit Equations Of Motion

We present the new, general, explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices. The systems may have holonomic and/or non-holonomic constraints, which may or may not satisfy D'Alembert's principle at each instant of time. The equation provides new insights into the behaviour of constrained motion and opens up new ways of modelling complex multi-body systems. Examples are provided and applications of the equation to such systems are illustrated.

Modeling and simulation of planar flexible link manipulators with rigid tip connections to revolute joints

Robotica ◽  
2004 ◽  
Vol 22 (3) ◽  
pp. 285-300 ◽  
Author(s):  
S. M. Megahed ◽  
K. T. Hamza
Keyword(s):  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Link ◽  
Link Length ◽  
Finite Segment ◽  
Revolute Joints ◽  
Body Dynamics ◽  
Lumped Masses ◽  
Multi Body ◽  
Rigid Zones

This paper presents the basis of a mathematical model for simulation of planar flexible-link manipulators, taking into consideration the effect of higher stiffness zones at the link tips. The proposed formulation is a variation of the finite segment multi-body dynamics approach. The formulation employs a consistent mass matrix in order to provide better approximation than the traditional lumped masses often encountered in the finite segment approach. The formulation is implemented into a computational code and tested through three examples; cantilever beam, rotating beam and three-link manipulator. In these examples, the length of the rigid tips at both sides of each link ranges from 0% to 6.25% of the whole link length. The zones of higher stiffness at the link tips are treated as short rigid zones. The effect of the rigid zones is averaged along with some portions of the flexible links, thereby allowing further simplification of the dynamic equations of motion. The simulation results demonstrate the effectiveness of the proposed modeling technique and show the importance of not ignoring the effect of the rigid tips.

A Moving Frame Method for Multi-Body Dynamics

2013 ◽  
Author(s):  
H. Murakami
Keyword(s):  
Equations Of Motion ◽  
Rigid Bodies ◽  
Explicit Representation ◽  
Moving Frame ◽  
Moving Frames ◽  
Body Dynamics ◽  
Moving Frame Method ◽  
Frame Method ◽  
Multi Body ◽  
Kinematics And Kinetics

Élie Cartan’s moving frame method, developed in differential geometry, has been applied to multi-body dynamics to derive equations of motion. The explicit representation of a body-attached orthonormal coordinate basis and its origin, referred to as a moving frame, enables the usage of the special orthogonal group, SO(3), and the special Euclidean group, SE(3), to describe kinematics and kinetics of interconnected bodies by joints and force elements. The moving frame representation using Theodore Frankel’s compact notation is adopted to alleviate theoretical complexities of the Lie group theory to which SO(3) and SE(3) belong. For the variational formulation, the restricted variation of angular velocity is derived for the moving frame method. Starting from two connected rigid bodies, it will be demonstrated that the explicit representation of moving frames renders straight-forward symbolic computations of three-dimensional kinematics and kinetics. This simplicity eliminates errors in computing analytical expressions for kinematic and kinetic variables and streamlines the coding effort for numerical solution. For controller design, if the degrees-of-freedom is small, the moving frame method allows a straight-forward derivation of equations of motion in analytical form.

Modeling and Simulation of Motorcycle Ride Comfort Based on Bump Road

2010 ◽  
Vol 139-141 ◽  
pp. 2643-2647 ◽  
Author(s):  
Dong Mei Yuan ◽  
Xiao Mei Zheng ◽  
Ying Yang
Keyword(s):  
Degrees Of Freedom ◽  
Ride Comfort ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Vertical Displacement ◽  
Dynamics Model ◽  
Body Dynamics ◽  
Space Formulation ◽  
Simulation Results ◽  
Multi Body

Through analyzing the motion when motorcycle runs on the bump road, the 5-DOF multi-body dynamics model of motorcycle is developed, the degrees of freedom include vertical displacement of sprung mass, rotation of sprung mass, vertical displacement of driver, and vertical displacement of front and rear suspension under sprung mass. According to Lagrange Equation, the differential equations of motion and state-space formulation are derived. Then bump road is simulated by triangle bump, and input displacement is programmed by MATLAB. With the input of bump road, motorcycle ride comfort is simulated, and the simulation results are verified by experiment results combined with two channels tire-coupling road simulator. It indicates that the simulation results and experiment results match well; the 5-DOF model has guidance for development of motorcycle ride comfort.

Analysis of the dynamic contact between rotating spur gears by finite element and multi-body dynamics techniques

2001 ◽  
Vol 215 (4) ◽  
pp. 423-435 ◽  
Author(s):  
K Lee
Keyword(s):  
Finite Element ◽  
Contact Point ◽  
Equations Of Motion ◽  
Mass Effect ◽  
Dynamic Contact ◽  
Spur Gears ◽  
Gear Teeth ◽  
Body Dynamics ◽  
Tooth Surfaces ◽  
Multi Body

A numerical method is presented for the dynamic contact analysis of spur gears rotating with very high angular speeds. For each gear an elastic tooth of distributed mass is connected to a rigid disc with kinematic constraints, and finite element formulations are used for the equations of motion of the teeth. The velocity and acceleration as well as the position of the contact point sliding on the mating gear teeth are precisely computed by simultaneously using the motions of a pair of rotating tooth surfaces. The equations of motion subjected to the kinematic constraint and contact condition are solved by enforcing the velocity and acceleration constraints as well as the displacement constraint. In the numerical simulation the importance of the mass effect of gear teeth is demonstrated, and it is shown that the solution is obtained even if gears repeat contact and separation.

An efficient hybrid method for dynamic interaction of train–track–bridge coupled system

2020 ◽  
Vol 47 (9) ◽  
pp. 1084-1093 ◽  
Author(s):  
Zhihui Zhu ◽  
Lei Zhang ◽  
Wei Gong ◽  
Lidong Wang ◽  
Yu Bai ◽  
...  
Keyword(s):  
Hybrid Method ◽  
Equations Of Motion ◽  
Coupled System ◽  
Superposition Method ◽  
Train Track ◽  
Mode Superposition Method ◽  
Body Dynamics ◽  
Direct Stiffness ◽  
Track Bridge ◽  
Multi Body

An efficient hybrid method (HM) is proposed by combining the direct stiffness method (DSM) and the mode superposition method (MSM) for analyzing the train–track–bridge coupled system (TTBS). The train and the track are modeled by applying the multi-body dynamics and the DSM, respectively. The bridge is modeled by applying the MSM that is efficient in capturing the dynamic behavior with a small number of modes. The train–track subsystem and the bridge subsystem are coupled by the interaction forces between them. The computational efficiency is significantly improved because of the considerably reduced number of equations of motion of the TTBS. Numerical simulations of a train traversing an arch railway bridge are performed and the results are compared with the field test data and the data from other methods, demonstrating the efficiency and accuracy of the proposed method.

Explicit Poincaré equations of motion for general constrained systems. Part II. Applications to multi-body dynamics and nonlinear control

2007 ◽  
Vol 463 (2082) ◽  
pp. 1435-1446 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Control ◽  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Constrained Systems ◽  
The Third ◽  
Highly Nonlinear ◽  
Body Dynamics ◽  
Multi Body

The power of the new equations of motion developed in part I of this paper is illustrated using three examples from multi-body dynamics. The first two examples deal with the problem of accurately controlling the orientation of a rigid body, while the third example deals with the synchronization of two rigid bodies so that their relative orientations are ‘locked’ through prescribed dynamical relationships. The ease, simplicity and accuracy with which control of such highly nonlinear systems is achieved are demonstrated.

New Approach to the Modeling of Complex Multibody Dynamical Systems

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Aaron Schutte ◽  
Firdaus Udwadia
Keyword(s):  
Multibody Systems ◽  
Multibody System ◽  
Equations Of Motion ◽  
Multiple Time ◽  
New Approach ◽  
Constrained Mechanical Systems ◽  
Step Procedure ◽  
Highly Nonlinear ◽  
Mass Matrices ◽  
General Complex

In this paper, a general method for modeling complex multibody systems is presented. The method utilizes recent results in analytical dynamics adapted to general complex multibody systems. The term complex is employed to denote those multibody systems whose equations of motion are highly nonlinear, nonautonomous, and possibly yield motions at multiple time and distance scales. These types of problems can easily become difficult to analyze because of the complexity of the equations of motion, which may grow rapidly as the number of component bodies in the multibody system increases. The approach considered herein simplifies the effort required in modeling general multibody systems by explicitly developing closed form expressions in terms of any desirable number of generalized coordinates that may appropriately describe the configuration of the multibody system. Furthermore, the approach is simple in implementation because it poses no restrictions on the total number and nature of modeling constraints used to construct the equations of motion of the multibody system. Conceptually, the method relies on a simple three-step procedure. It utilizes the Udwadia–Phohomsiri equation, which describes the explicit equations of motion for constrained mechanical systems with singular mass matrices. The simplicity of the method and its accuracy is illustrated by modeling a multibody spacecraft system.

Coupled computational fluid and multi-body dynamics suspension boat modeling

2017 ◽  
Vol 24 (18) ◽  
pp. 4260-4281 ◽  
Author(s):  
Maysam Mousaviraad ◽  
Michael Conger ◽  
Shanti Bhushan ◽  
Frederick Stern ◽  
Andrew Peterson ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
Ride Quality ◽  
Immersed Boundary ◽  
Dynamics Modeling ◽  
Dynamic Interactions ◽  
Coupled Equations ◽  
Body Dynamics ◽  
Calm Water ◽  
Vertical Accelerations ◽  
Multi Body

Multiphysics modeling, code development, and validation by full-scale experiments is presented for hydrodynamic/suspension-dynamic interactions of a novel ocean vehicle, the Wave Adaptive Modular Vessel (WAM-V). The boat is a pontoon catamaran with hinged engine pods and elevated payload supported by suspension and articulation systems. Computational fluid dynamics models specific to WAM-V are developed which include hinged pod dynamics, water-jet propulsion modeling, and immersed boundary method for flow in the gap between pontoon and pod. Multi-body dynamics modeling for the suspension and upper-structure dynamic is developed in MATLAB Simulink. Coupled equations of motion are developed and solved iteratively through either one-way or two-way coupling methods to converge on flow-field, pontoon motions, pod motions, waterjet forces, and suspension motions. Validation experiments include cylinder drop with suspended mass and 33-feet WAM-V sea-trials in calm water and waves. Computational results show that two-way coupling is necessary to capture the physics of the interactions. The experimental trends are predicted well and errors are mostly comparable to those for rigid boats, however, in some cases the errors are larger, which is expected due to the complexity of the current studies. Ride quality analyses show that WAM-V suspension is effective in reducing payload vertical accelerations in waves by 73% compared to the same boat with rigid upper-structure.

A Numerical and Experimental Analysis of a Chain of Flexible Bodies

1994 ◽  
Vol 116 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Marco Giovagnoni
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Numerical Data ◽  
Small Displacement ◽  
Flexible Bodies ◽  
Rigid Link ◽  
Body Dynamics ◽  
A Chain ◽  
Multi Body ◽  
Kinematic Solution

A flexible multi-body dynamics approach is described. It uses an equivalent rigid link system from which are measured small displacements. The equations of motion are obtained by direct application of the principle of virtual work. Some terms in the virtual and real components have been neglected by virtue of the small displacement assumption. The use of sensitivity coefficients allows one to obtain a formulation which can be easily interfaced with any kinematic solution algorithm. It also enables one to check the correctness of the chosen equivalent rigid link system. The theory is then employed to reproduce numerically the experimental recordings obtained from a flexible linkage. Agreement between experimental and numerical data is good.

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