Dynamics of Flexible Multibody Systems Using Bond Graphs and Lagrange Multipliers

1990 ◽  
Vol 112 (1) ◽  
pp. 30-35 ◽  
Author(s):  
B. Samanta
Keyword(s):  
Rigid Body ◽  
Lagrange Multipliers ◽  
Equations Of Motion ◽  
Constrained Systems ◽  
System Response ◽  
Bond Graphs ◽  
Higher Derivatives ◽  
Model Constraint ◽  
System Equations ◽  
Derivatives Of

A procedure is presented to study the dynamics of interconnected flexible systems using bond graphs. The concept of Lagrange multipliers, which are commonly used in analysis of constrained systems, is introduced in the procedure. The overall motions of each of the component bodies are described in terms of large rigid body motions and small elastic vibrations. Bond graphs are used to represent both rigid body and flexible dynamics of each body in a unified manner. Bond graphs of such sub-systems are coupled to one another satisfying the kinematic constraints at the interfaces to get the overall system model. Constraint reactions are introduced in the form of Lagrange multipliers at the interfaces without disturbing the integral causality in the subsystem models, which leads to easy derivation of system equations. The equations of motion and higher derivatives of the constraint relations are integrated to obtain the constraint reactions and the system response. The procedure is illustrated by an example system and results are in good agreement with those presented earlier.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

Dynamic Response of Spinning Blades Subjected to Gyroscopic Motion

1994 ◽  
Vol 116 (1) ◽  
pp. 6-15 ◽  
Author(s):  
T. H. Young ◽  
G. T. Liou
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Spin Axis ◽  
System Response ◽  
First Order ◽  
System Equations ◽  
Element Technique ◽  
Gyroscopic Motion

This paper presents an investigation into the vibration and stability of a blade spinning with respect to a nonfixed axis. Due to the motion of the spin axis, parametric instability of the blade may occur in certain situations. In this work, the discretized equations of motion are first formulated by the finite element technique. Then the system equations are transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations. Each set of differential equations is solved analytically by the method of multiple scales if the precessional speed of the spin axis is assumed to be small compared to the spin rate of the blade, yielding the system response and the expressions for the boundaries of the unstable regions. Finally, the effects of system parameters on the changes in these boundaries are studied numerically.

Dynamic Modeling and Simulation of Flexible Robots With Prismatic Joints

1990 ◽  
Vol 112 (3) ◽  
pp. 307-314 ◽  
Author(s):  
Ye-Chen Pan ◽  
R. A. Scott ◽  
A. Galip Ulsoy
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Numerical Procedure ◽  
Differential Algebraic Equations ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Integration Scheme ◽  
System Response ◽  
Constraint Equations ◽  
Prismatic Joints

A dynamic model for flexible manipulators with prismatic joints is presented in Part I of this study. Floating frames following a nominal rigid body motion are introduced to describe the kinematics of the flexible links. A Lagrangian approach is used in deriving the equations of motion. The work done by the rigid body axial force through the axial shortening of the link due to transverse deformations is included in the Lagrangian function. Kinematic constraint equations are used to describe the compatibility conditions associated with revolute joints and prismatic joints, and incorporated into the equations of motion by Lagrange multipliers. The small displacements due to the flexibility of the links are then discretized by a displacement based finite element method. Equations of motion are derived for the cases of prescribed rigid body motion as well as prescribed joint torques/forces through application of Lagrange’s equations. The equations of motion and the constraint equations result in a set of differential algebraic equations. A numerical procedure combining a constraint stabilization method and a Newmark direct integration scheme is then applied to obtain the system response. An example, previously treated in the literature, is presented to validate the modeling and solution methods used in this study.

A Modal Analysis Solution Technique to the Equations of Motion for Elastic Mechanism Systems Including the Rigid-Body and Elastic Motion Coupling Terms

1993 ◽  
Vol 115 (2) ◽  
pp. 314-323 ◽  
Author(s):  
A. G. Jablokow ◽  
S. Nagarajan ◽  
D. A. Turcic
Keyword(s):  
Rigid Body ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Solution Technique ◽  
System Response ◽  
Matrix Equations ◽  
Analysis Techniques ◽  
The Matrix ◽  
Elastic Mechanism ◽  
Coupling Terms

This paper presents a modal analysis solution technique for the matrix equations of motion of elastic mechanism systems including the rigid-body elastic motion coupling terms and general damping. In many cases, researchers have neglected these terms because they complicate and diminish the efficiency of the solution process. This has been justified by assuming that the terms are small and do not affect the system response. The results obtained using the techniques developed within show this is true for some, but not all mechanisms. The solution technique adds the rigid-body elastic motion coupling terms to the system mass, damping, and stiffness matrices thus allowing them to affect the system response appropriately. The resulting nonsymmetric matrices are then rewritten in first order form allowing general damping to be included in the analysis. Modal analysis techniques are utilized to solve for the steady-state response of the elastic mechanism system. Thus the methods developed in this work provide a technique for including the rigid-body elastic motion coupling terms and general damping in the equations of motion while maintaining the advantage of using efficient modal analysis techniques of finding the response of the system. A number of examples are presented that establish the validity of this approach to the solution of the matrix equations of motion for elastic mechanism systems. The results show that the rigid-body elastic motion coupling terms can become significant at higher operating speeds.

A note on the Hamiltonian theory of quantization. II

1947 ◽  
Vol 43 (2) ◽  
pp. 196-204 ◽  
Author(s):  
T. S. Chang
Keyword(s):  
Variational Principles ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Field Equations ◽  
Gauge Invariant ◽  
Commutation Relations ◽  
Hamiltonian Theory ◽  
Higher Derivatives ◽  
Missing Momenta ◽  
Derivatives Of

It is pointed out that the equations of motion for any field obtained by varying a Lagrangian subject to auxiliary conditions are exactly equivalent to a certain set of canonical equations and that the commutation relations between the dynamical variables for the latter equations are Lorentz-invariant. By extending the theory to Lagrangians containing higher derivatives of the field quantities, it is shown that any given set of field equations can be put into the canonical form, though it is not derived from variational principles. The question of Lagrangians with missing momenta is also considered. It is shown that if the Lagrangian is ‘gauge-invariant’, some of the p's must be missing and the corresponding Eulerian equations can be replaced by equations containing no q and then can be replaced by initial conditions. The commutation relations between gauge-invariant quantities are Lorentz-invariant. For Lagrangians which are not gauge-invariant but are such as to have missing momenta, the passage to quantum theory will in general give rise to non-Lorentz-invariant commutation relations. In both cases, the equations of motion can be cast in canonical forms.

Modeling of Multibody Flexible Articulated Structures With Mutually Coupled Motions: Part I — General Theory

1992 ◽  
Author(s):  
Jiechi Xu ◽  
Joseph R. Baumgarten
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Local Level ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Moving Frame ◽  
Element Analysis ◽  
Highly Nonlinear ◽  
Geometric Boundary ◽  
System Equations

Abstract In the present paper a general systematic modeling procedure has been conducted in deriving dynamic equations of motion using Lagrange’s approach for a spatial multibody structural system involving rigid bodies and elastic members. Both the rigid body degrees of freedom and the elastic degrees of freedom are considered as unknown generalized coordinates of the entire system in order to reflect the nature of mutually coupled rigid body and elastic motions. The assumption of specified rigid body gross motion is no longer necessary in the equation derivation and the resulting differential equations are highly nonlinear. Finite element analysis (FEA) with direct stiffness method has been employed to model the flexible substructures. Nonlinear coupling terms between the rigid body and elastic motions are fully derived and are explicitly expressed in matrix form. The equations of motion of each individual subsystem are formulated based on a moving frame instead of a traditional inertial frame. These local level equations of motion are assembled to obtain the system equations with the implementation of geometric boundary conditions by means of a compatibility matrix.

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.

scholarly journals New non-perturbative de Sitter vacua in α′-complete cosmology

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Carmen A. Núñez ◽  
Facundo Emanuel Rost
Keyword(s):  
Equations Of Motion ◽  
Generalized Equations ◽  
De Sitter ◽  
Scale Factors ◽  
Spatial Dimensions ◽  
Higher Derivatives ◽  
De Sitter Vacua ◽  
The Stability ◽  
Derivatives Of ◽  
Constant Dilaton

Abstract The α′-complete cosmology developed by Hohm and Zwiebach classifies the O(d, d; ℝ) invariant theories involving metric, b-field and dilaton that only depend on time, to all orders in α′. Some of these theories feature non-perturbative isotropic de Sitter vacua in the string frame, generated by the infinite number of higher-derivatives of O(d, d; ℝ) multiplets. Extending the isotropic ansatz, we construct stable and unstable non-perturbative de Sitter solutions in the string and Einstein frames. The generalized equations of motion admit new solutions, including anisotropic d + 1-dimensional metrics and non-vanishing b-field. In particular, we find dSn+1× Td−n geometries with constant dilaton, and also metrics with bounded scale factors in the spatial dimensions with non-trivial b-field. We discuss the stability and non-perturbative character of the solutions, as well as possible applications.

Nonlinear Interactions in Systems of Multiple Order Centrifugal Pendulum Vibration Absorbers

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Brendan J. Vidmar ◽  
Steven W. Shaw ◽  
Brian F. Feeny ◽  
Bruce K. Geist
Keyword(s):  
Equations Of Motion ◽  
System Stability ◽  
System Response ◽  
Nonlinear Interactions ◽  
Vibration Absorbers ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
System Equations ◽  
Improved Performance ◽  
Nonlinear Equations Of Motion ◽  
Selection Of

We consider nonlinear interactions in systems of order-tuned torsional vibration absorbers with sets of absorbers tuned to different orders. In all current applications, absorber systems are designed to reduce torsional vibrations at a single order. However, when two or more excitation orders are present and absorbers are introduced to address different orders, nonlinear interactions become possible under certain resonance conditions. Under these conditions, a common example of which occurs for orders n and 2n, crosstalk between the absorbers, acting through the rotor inertia, can result in instabilities that are detrimental to system response. In order to design absorber systems that avoid these interactions, and to explore possible improved performance with sets of absorbers tuned to different orders, we develop predictive models that allow one to examine the effects of absorber mass distribution and tuning. These models are based on perturbation methods applied to the system equations of motion, and they yield system response features, including absorber and rotor response amplitudes and stability, as a function of parameters of interest. The model-based analytical results are compared against numerical simulations of the complete nonlinear equations of motion, and are shown to be in good agreement. These results are useful for the selection of absorber parameters to achieve desired performance. For example, they allow for approximate closed form expressions for the ratio of absorber masses at the two orders that yield optimal performance. It is also found that utilizing multiple order absorber systems can be beneficial for system stability, even when only a single excitation order is present.

Analysis and Design of Multiple Order Centrifugal Pendulum Vibration Absorbers

2012 ◽  
Author(s):  
Brendan J. Vidmar ◽  
Steven W. Shaw ◽  
Brian F. Feeny ◽  
Bruce K. Geist
Keyword(s):  
Equations Of Motion ◽  
System Response ◽  
Torsional Vibrations ◽  
Nonlinear Interactions ◽  
Vibration Absorbers ◽  
Analysis And Design ◽  
Engine Order ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
System Equations ◽  
Nonlinear Equations Of Motion

We consider nonlinear interactions in systems of order-tuned torsional vibration absorbers. These absorbers, which consist of centrifugally driven pendulums fitted to a rotor, are used to reduce engine-order torsional vibrations in rotating machines, including automotive engines, helicopter rotors, and light aircraft engines. In all current applications, absorber systems are designed to reduce torsional vibrations at a single order. However, when two or more excitation orders are present and absorbers are introduced to address different orders, undesirable nonlinear interactions become possible under certain resonance conditions. Under these conditions, a common example of which occurs for orders n and 2n, crosstalk between the absorbers, acting through the rotor inertia, can result in instabilities that are detrimental to system response. In order to design absorber systems that avoid these interactions, we develop predictive models that allow one to select proper tuning and sizing of the absorbers. These models are based on perturbation methods applied to the system equations of motion, and they yield system response features, including absorber and rotor response amplitudes and stability, as a function of parameters of interest. The model-based analytical results are compared against numerical simulations of the complete nonlinear equations of motion, and are shown to be in good agreement. These results are useful for the selection of absorber parameters for desired performance. For example, they allow for approximate closed form expressions for the ratio of absorber masses at the two orders that yield optimal performance.

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