A Closed-Form Formalism for Controlled and Hybrid Rigid/Elastic Multibody Systems: Part II — Symbolic Implementation and Applications

1991 ◽  
Author(s):  
Junghsen Lieh ◽  
Imitiaz Haque
Keyword(s):  
Closed Form ◽  
Equations Of Motion ◽  
Structural Flexibility ◽  
Symbolic Language ◽  
Structural Vibrations ◽  
Optimal Control Strategy ◽  
Active Suspensions ◽  
Elastic Multibody Systems ◽  
System Order ◽  
Crank Mechanism

Abstract A program is developed on a DECstation using the symbolic language MAPLE which generates the equations of motion in a closed form and reduces the system order symbolically. A procedure that can make symbolic simplification and linearization is provided. The integration of shape functions is performed symbolically. Both nonlinear and linearized equations of motion with control are established in FORTRAN format. Several models including an elastic vehicle with active suspensions, an elastic robotic manipulator and an elastic slider-crank mechanism with both joint and structural flexibility are generated. Numerical simulation for the active vehicle model using an optimal control strategy is presented. The effect of active suspensions on vehicle and structural vibrations is briefly discussed. A comparison between the nonlinear and linearized robot models is given. Simulation results of the slider-crank mechanism are also presented.

A Closed-Form Formalism for Controlled and Hybrid Rigid/Elastic Multibody Systems: Part I — Formulation

1991 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz Haque
Keyword(s):  
Closed Form ◽  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Rigid Body Dynamics ◽  
New Formalism ◽  
Assumed Mode Method ◽  
Elastic Multibody Systems ◽  
Moving Base ◽  
Generalized Force

Abstract A new formalism leading to closed-form formulation of equations for controlled elastic multibody systems is presented. The method is derived from the virtual work principle and includes the effects of a moving base and rigid body dynamics. The elastic members are treated as Euler-Bernoulli beams and the assumed-mode method is adopted. The equations of motion are expanded in a closed form with a minimum set of variables using the generalized coordinate partitioning and a Jacobian matrix expansion. The inertia matrix, nonlinear coupling vector, generalized force vector and other terms containing the velocity and acceleration effects of a moving base are formulated separately. The formalism facilitates matrix computations and is very suitable for symbolic processing. The method is very systematic and general and can be applied to a multibody system subject to control and constraint conditions. Derivation of the formalism is presented in part I of the article, and symbolic implementation and application of the formalism to various elastic mechanical systems are presented in part II.

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.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

Modeling and Dynamic Analysis of a Slider-Crank Mechanism With Flexible Coupler and Joint

1991 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz Haque
Keyword(s):  
Dynamic Analysis ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Joint Stiffness ◽  
Mechanism Model ◽  
Double Frequency ◽  
Varying Coefficients ◽  
Matrix Expansion ◽  
Time Varying Coefficients ◽  
Crank Mechanism

Abstract Modeling and dynamic analysis of a slider-crank mechanism with flexible joint and coupler is presented. The equations of motion of the mechanism model are formulated using a virtual work multibody formalism and cast in terms of a minimum set of generalized coordinates through a Jacobian matrix expansion. Numerical results show the influence of time-varying coefficients on the mechanism dynamic behavior due to a repeated task. The results illustrate that the joint motion and coupler deformation are highly coupled. The joint response is dominated by double frequency of input, however, the coupler deformation is influenced by the same frequency as that of excitation. Increase in joint stiffness tends to decrease the variations in coupler deformation.

Investigation of Parametric Resonance Instability in a Flexible Rod Slider Crank Mechanism

1991 ◽  
Author(s):  
David G. Beale ◽  
Shyr-Wen Lee
Keyword(s):  
Potential Energy ◽  
Parametric Resonance ◽  
Variational Approach ◽  
Equations Of Motion ◽  
Monodromy Matrix ◽  
Bending Energy ◽  
Constrained Equations ◽  
Beam Bending ◽  
Floating Frame ◽  
Crank Mechanism

Abstract A direct variational approach with a floating frame is presented to derive the ordinary differential equations of motion of a flexible rod, constant crank speed slider crank mechanism. Potential energy terms contained in the derivation include beam bending energy and energy in foreshortening of the rod tip (which were selected because of the importance of these terms in a pinned-pinned rod parametric resonance). A symbolic manipulator code is used to reduce the constrained equations of motion to unconstrained nonlinear equations. A linearized version of these equations is used to explore parametric resonance stability-instability zones at low crank speeds and small deflections by a monodromy matrix technique.

An Approach for the Dynamics of Robotic Manipulators With Structural Flexibility of Links and Joints

1997 ◽  
Author(s):  
J. Kövecses ◽  
R. G. Fenton ◽  
W. L. Cleghorn
Keyword(s):  
Equations Of Motion ◽  
Robotic Manipulators ◽  
Structural Flexibility ◽  
Flexible Link ◽  
Dynamic Effects ◽  
Flexible Joint ◽  
Modeling And Analysis ◽  
Discrete Parameter ◽  
Rigid Link ◽  
Different Types

Abstract In this paper, an approach is presented for the dynamic modeling and analysis of robotic manipulators having structural flexibility in the links and joints. The formulation allows the user to include different types of flexibilities, as required. This approach includes the dynamic effects of joint driving systems by considering the mass and moments of inertia of their elements, the rotor-link interactions, and the gear reduction ratios; all of which can have significant influences on the behavior of the manipulator. Both distributed-discrete and discretized-discrete parameter models of a robot can be analysed. In the discretized-discrete case, dynamic equations of motion are developed for four model types: rigid link - rigid joint, rigid link - flexible joint, flexible link - rigid joint, and flexible link - flexible joint. An example of a two-link manipulator is considered. Simulation results are presented for different models (flexible joint - rigid link, rigid joint - flexible link, flexible joint - flexible link) of the manipulator. The computations show the influence of joint and link flexibilities on the manipulator performance.

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.

A Comfort Oriented Control Strategy for Semi-Active Suspensions Based on Half Car Model

2010 ◽  
Author(s):  
Cristiano Spelta ◽  
Diego Delvecchio ◽  
Sergio M. Savaresi
Keyword(s):  
Optimal Control ◽  
Control Strategy ◽  
Optimal Performance ◽  
Optimal Control Strategy ◽  
Active Suspensions ◽  
Quarter Car Model ◽  
Car Model ◽  
Control Goal ◽  
Quarter Car ◽  
Effective Description

This paper is devoted to the design of a novel semi-active comfort-oriented control strategy based on the “half-car” modeling of the vehicle. The half car model is an effective description of the vertical behaviors in a vehicle like a motorcycle, since it is able to represent both the heave and pitch dynamics. A recent control strategy (the “Mix-1-Sensor”) have been proven to be the quasi-optimal control strategy when the system is described with a quarter car model and the comfort objective is the control goal. This paper presents an analysis of the performances of the Mix-1-Sensor implemented in a half car: this strategy is able to guarantee a quasi optimal performance in terms of heave dynamics but it is not able to manage the pitch dynamics efficiently. A pitch-oriented extension of this strategy is proposed in order to guarantee a better filtering of the pitch dynamics.

Automating the Derivation of the Equations of Motion of a Multibody Dynamic System With Uncertainty Using Polynomial Chaos Theory and Variational Work

2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Paul S. Ryan ◽  
Sarah C. Baxter ◽  
Philip A. Voglewede
Keyword(s):  
Chaos Theory ◽  
Polynomial Chaos ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Multibody Dynamic ◽  
Mass Spring ◽  
Polynomial Chaos Theory ◽  
Multibody Dynamic System ◽  
Crank Mechanism

Abstract Understanding how variation impacts a multibody dynamic (MBD) system's response is important to ensure the robustness of a system. However, how the variation propagates into the MBD system is complicated because MBD systems are typically governed by a system of large differential algebraic equations. This paper presents a novel process, variational work, along with the polynomial chaos multibody dynamics (PCMBoD) automation process for utilizing polynomial chaos theory (PCT) in the analysis of uncertainties in an MBD system. Variational work allows the complexity of the traditional PCT approach to be reduced. With variational work and the constrained Lagrangian formulation, the equations of motion of an MBD PCT system can be constructed using the PCMBoD automated process. To demonstrate the PCMBoD process, two examples, a mass-spring-damper and a two link slider–crank mechanism, are shown.

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