Dynamical Equations With Non-Ideal Constraints: New and Old Alternatives

2007 ◽  
Author(s):  
Arun K. Banerjee ◽  
Mark Lemak
Keyword(s):  
Equations Of Motion ◽  
Alternative Form ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Constraint Force ◽  
Kane’S Method ◽  
Kane's Method ◽  
Ellipsoidal Surface ◽  
Ideal Constraint ◽  
The Ideal

This paper deals with the motion of mechanical systems with non-ideal constraints, defined as constraints where the forces associated with the constraint do work. The first objective of the paper is to show that two newly published formulations of equations of motion of systems with such non-ideal constraints are unnecessarily complex for situations where the non-ideal constraint force does not depend on the ideal constraint force, because they introduce and then eliminate these non-working constraint forces. We point out that a method already exists for nonideal constraints, namely, Kane’s equations, which are simpler because, among other things, they are based on automatic elimination of non-working constraints. The examples considered in these recent publications are worked out with Kane’s method to show the applicability and simplicity of Kane’s method for non-ideal constraints. A second objective of the paper is to present an alternative form of equations for systems where the non-ideal constraint force depends on the ideal constraint force, as in the case of Coulomb friction. The formulation is shown to lend itself naturally to also analyzing impact dynamics. The method is applied to the dynamics of a slug moving against friction on a moving ellipsoidal surface. Such a crude model may simulate, in essence, propellant motion in a tank in zero-g, or during docking of a spacecraft.

scholarly journals An Argument Against Augmenting the Lagrangean for Nonholonomic Systems

2009 ◽  
Vol 76 (3) ◽  
Author(s):  
Carlos M. Roithmayr ◽  
Dewey H. Hodges
Keyword(s):  
Nonholonomic System ◽  
Nonholonomic Constraint ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Euler Method ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
New Approach ◽  
Constraint Equations ◽  
Kane's Method

Although it is known that correct dynamical equations of motion for a nonholonomic system cannot be obtained from a Lagrangean that has been augmented with a sum of the nonholonomic constraint equations weighted with multipliers, previous publications suggest otherwise. One published example that was proposed in support of augmentation purportedly demonstrates that an accepted method fails to produce correct equations of motion whereas augmentation leads to correct equations. This present paper shows that, in fact, the opposite is true. The correct equations, previously discounted on the basis of a flawed application of the Newton–Euler method, are verified by using Kane’s method together with a new approach for determining the directions of constraint forces.

Dynamics Modeling of Elastic Multibody Systems Using Kane’s Method As Applied to a Flexible Robot

1997 ◽  
Author(s):  
Ali Meghdari ◽  
Farbod Fahimi
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Component Mode Synthesis ◽  
Flexible Robot ◽  
Dynamics Modeling ◽  
Generalized Coordinates ◽  
Kane’S Method ◽  
Kane's Method ◽  
Elastic Multibody Systems ◽  
Elastic Form

Abstract Generalization of Kane’s equations of motion for elastic multibody systems is considered. Initially, finite element techniques are used to generate the elastic form of generalized coordinates. Then, the number of elastic coordinates are reduced by the component mode synthesis. Finally, Kane’s method is applied to obtain the equations of motion of such systems. Using this method, dynamic model of an elastic robot with one degree of freedom is presented.

Comparison of Kane’s and Lagrange’s Methods in Analysis of Constrained Dynamical Systems

Robotica ◽  
2020 ◽  
Vol 38 (12) ◽  
pp. 2138-2150
Author(s):  
Amin Talaeizadeh ◽  
Mahmoodreza Forootan ◽  
Mehdi Zabihi ◽  
Hossein Nejat Pishkenari
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Constrained Systems ◽  
Robot Dynamics ◽  
Energy Error ◽  
Kane’S Method ◽  
Kane's Method ◽  
Constrained Dynamical Systems ◽  
Error Energy

SUMMARYDynamic modeling is a fundamental step in analyzing the movement of any mechanical system. Methods for dynamical modeling of constrained systems have been widely developed to improve the accuracy and minimize computational cost during simulations. The necessity to satisfy constraint equations as well as the equations of motion makes it more critical to use numerical techniques that are successful in decreasing the number of computational operations and numerical errors for complex dynamical systems. In this study, performance of a variant of Kane’s method compared to six different techniques based on the Lagrange’s equations is shown. To evaluate the performance of the mentioned methods, snake-like robot dynamics is considered and different aspects such as the number of the most time-consuming computational operations, constraint error, energy error, and CPU time assigned to each method are compared. The simulation results demonstrate the superiority of the variant of Kane’s method concerning the other ones.

Inverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual work

Robotica ◽  
2009 ◽  
Vol 27 (2) ◽  
pp. 259-268 ◽  
Author(s):  
Yongjie Zhao ◽  
Feng Gao
Keyword(s):  
Parallel Manipulator ◽  
Inverse Dynamics ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Robot System ◽  
Lead Screw ◽  
Moving Platform

SUMMARYIn this paper, the inverse dynamics of the 6-dof out-parallel manipulator is formulated by means of the principle of virtual work and the concept of link Jacobian matrices. The dynamical equations of motion include the rotation inertia of motor–coupler–screw and the term caused by the external force and moment exerted at the moving platform. The approach described here leads to efficient algorithms since the constraint forces and moments of the robot system have been eliminated from the equations of motion and there is no differential equation for the whole procedure. Numerical simulation for the inverse dynamics of a 6-dof out-parallel manipulator is illustrated. The whole actuating torques and the torques caused by gravity, velocity, acceleration, moving platform, strut, carriage, and the rotation inertia of the lead screw, motor rotor and coupler have been computed.

DYNAMICS OF FLEXIBLE MULTIBODY MECHANICAL SYSTEMS

1991 ◽  
Vol 15 (3) ◽  
pp. 235-256 ◽  
Author(s):  
X. Cyril ◽  
J. Angeles ◽  
A. Misra
Keyword(s):  
Equations Of Motion ◽  
Matrix Multiplication ◽  
Mechanical Systems ◽  
Dynamical Behaviour ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Simple Matrix ◽  
Kinematic Velocity ◽  
Velocity Constraints ◽  
The Individual

In this paper the formulation and simulation of the dynamical equations of multibody mechanical systems comprising of both rigid and flexible-links are accomplished in two steps: in the first step, each link is considered as an unconstrained body and hence, its Euler-Lagrange (EL) equations are derived disregarding the kinematic couplings; in the second step, the individual-link equations, along with the associated constraint forces, are assembled to obtain the constrained dynamical equations of the multibody system. These constraint forces are then efficiently eliminated by simple matrix multiplication of the said equations by the transpose of the natural orthogonal complement of kinematic velocity constraints to obtain the independent dynamical equations. The equations of motion are solved for the generalized accelerations using the Cholesky decomposition method and integrated using Gear’s method for stiff differential equations. Finally, the dynamical behaviour of the Shuttle Remote Manipulator when performing a typical manoeuvre is determined using the above approach.

Identifying Sets of Constraint Forces by Inspection

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Carlos M. Roithmayr ◽  
Dewey H. Hodges
Keyword(s):  
Mechanical System ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Vector Form ◽  
Constraint Forces ◽  
Constraint Force ◽  
Symbolic Algebra ◽  
Constraint Equations ◽  
Angular Velocities

A mechanical system is often modeled as a set of particles and rigid bodies, some of which are constrained in one way or another. A concise method is proposed for identifying a set of constraint forces needed to ensure the restrictions are met. Identification consists of determining the direction of each constraint force and the point at which it must be applied, as well as the direction of the torque of each constraint force couple, together with the body on which the couple acts. This important information can be determined simply by inspecting constraint equations written in vector form. For the kinds of constraints commonly encountered, the constraint equations are expressed in terms of dot products involving velocities of the affected points or particles and angular velocities of the bodies concerned. The technique of expressing constraint equations in vector form and identifying constraint forces by inspection is useful when one is deriving explicit, analytical equations of motion by hand or with the aid of symbolic algebra software, as demonstrated with several examples.

A geometrical interpretation of Kane’s Equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 69-87 ◽  
Keyword(s):  
Tangent Space ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Analytical Mechanics ◽  
Kane’S Method ◽  
Kane's Method ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Spanning Set ◽  
Geometric Picture

The method for the development of the equations of motion for systems of constrained particles and rigid bodies, developed by T. R. Kane and called Kane’s Equations, is discussed from a geometric viewpoint. It is shown that what Kane calls partial velocities and partial angular velocities may be interpreted as components of tangent vectors to the system’s configuration manifold. The geometric picture, when attached to Kane’s formalism shows that Kane’s Equations are projections of the Newton-Euler equations of motion onto a spanning set of the configuration manifold’s tangent space. One advantage of Kane’s method, is that both non-holonomic and non-conservative systems are easily included in the same formalism. This easily follows from the geometry. It is also shown that by transformation to an orthogonal spanning set, the equations can be diagonalized in terms of what Kane calls the generalized speeds. A further advantage of the geometric picture lies in the treatment of constraint forces which can be expanded in terms of a spanning set for the orthogonal complement of the configuration tangent space. In all these developments, explicit use is made of a concrete realization of the multidimensional vectors which are called K -vectors for a K -component system. It is argued that the current presentation also provides a clear tutorial route to Kane’s method for those schooled in classical analytical mechanics.

scholarly journals Motions of a Constrained Spherical Pendulum in an Arbitrarily Moving Reference Frame: The Automobile Seatbelt Inertial Sensor

1995 ◽  
Vol 2 (3) ◽  
pp. 227-236 ◽  
Author(s):  
A. Peter Allan ◽  
Miles A. Townsend
Keyword(s):  
Numerical Simulation ◽  
Reference Frame ◽  
Inertial Sensor ◽  
Equations Of Motion ◽  
Sensor Design ◽  
Spherical Pendulum ◽  
Crash Data ◽  
Kane’S Method ◽  
Kane's Method ◽  
Automobile Crash

A common automatic seatbelt inertial sensor design, comprised of a constrained spherical pendulum, is modeled to study its motions and possible unintentional release during vehicle emergency maneuvers. The kinematics are derived for the system with the most general inputs: arbitrary pivot motions. The influence of forces due to gravity and constraint torque functions is developed. The equations of motion are then derived using Kane's method. The equations of motion are used in a numerical simulation with both actual and hypothetical automobile crash data.

scholarly journals Use of Kane’s Method for Multi-Body Dynamic Modelling and Control of Spar-Type Floating Offshore Wind Turbines

Energies ◽  
2021 ◽  
Vol 14 (20) ◽  
pp. 6635
Author(s):  
Saptarshi Sarkar ◽  
Breiffni Fitzgerald
Keyword(s):  
Wind Turbine ◽  
Equations Of Motion ◽  
Dynamic Modelling ◽  
Offshore Wind ◽  
Offshore Wind Turbine ◽  
Kane’S Method ◽  
Kane's Method ◽  
Floating Offshore Wind Turbines ◽  
Multi Body ◽  
And Control

This paper demonstrates the use of Kane’s method to derive equations of motion for a spar-type floating offshore wind turbine taking into account the flexibility of the members. The recently emerged Kane’s method reduces the effort required to derive equations of motion for complex multi-body systems, making them simpler to model and more readily solved by computers. Further, the installation procedure of external vibration control devices on the wind turbine using Kane’s method is described, and the ease of using this method has been demonstrated. A tuned mass damper inerter (TMDI) is installed in the tower for illustration. The excellent vibration mitigation properties of the TMDI are also presented in this paper.

Sign in / Sign up

Export Citation Format

Share Document